Java基础快速入门：TreeSet底层数据结构——树

Wang's Blog2026-06-28 15:42

本文纲要

  1. 引言
  2. 树的基本结构：节点与二叉树
    2.1 节点的构成
    2.2 什么是二叉树
    2.3 树的基本术语
  3. 二叉查找树（二叉排序树）
    3.1 定义与特点
    3.2 添加节点过程
    3.3 代码演示：节点类与插入
  4. 平衡二叉树
    4.1 为什么需要平衡
    4.2 定义与判断
  5. 平衡二叉树的旋转机制
    5.1 左旋（Left Rotation）
    5.2 右旋（Right Rotation）
    5.3 左左、左右、右右、右左四种情况
  6. 总结

引言

在 Java 集合框架中，TreeSet 是一种有序的 Set 实现。它的底层数据结构是红黑树（一种自平衡的二叉查找树）。要深入理解 TreeSet 的工作原理，首先要掌握树结构的基础知识。

本文将从最基础的二叉树讲起，逐步过渡到二叉查找树、平衡二叉树，并详细讲解平衡二叉树维持平衡的核心操作------左旋与右旋。这些内容是理解 TreeMap、TreeSet 底层实现的关键。

树的基本结构：节点与二叉树

1 ） 节点的构成

在树结构中，每一个元素称为一个节点（Node）。每一个节点通常包含四部分信息：

父节点的地址

自身存储的值

左子节点的地址

右子节点的地址

可以用如下 Java 类来表示：

java 复制代码 
public class TreeNode {
   // 父节点 
   TreeNode parent;
   // 存储的值 
   int value;
   // 左子节点 
   TreeNode left;
   // 右子节点 
   TreeNode right;

   public TreeNode(int value) {
       this.value = value;
   }
}

一个节点如果没有父节点（即为根节点），则 parent 为 null；如果没有左子节点或右子节点，则相应的 left、right 为 null。

以下图为例：
#mermaid-svg-3IUxoHcixdkHh3nW{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;fill:#333;}@keyframes edge-animation-frame{from{stroke-dashoffset:0;}}@keyframes dash{to{stroke-dashoffset:0;}}#mermaid-svg-3IUxoHcixdkHh3nW .edge-animation-slow{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 50s linear infinite;stroke-linecap:round;}#mermaid-svg-3IUxoHcixdkHh3nW .edge-animation-fast{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 20s linear infinite;stroke-linecap:round;}#mermaid-svg-3IUxoHcixdkHh3nW .error-icon{fill:#552222;}#mermaid-svg-3IUxoHcixdkHh3nW .error-text{fill:#552222;stroke:#552222;}#mermaid-svg-3IUxoHcixdkHh3nW .edge-thickness-normal{stroke-width:1px;}#mermaid-svg-3IUxoHcixdkHh3nW .edge-thickness-thick{stroke-width:3.5px;}#mermaid-svg-3IUxoHcixdkHh3nW .edge-pattern-solid{stroke-dasharray:0;}#mermaid-svg-3IUxoHcixdkHh3nW .edge-thickness-invisible{stroke-width:0;fill:none;}#mermaid-svg-3IUxoHcixdkHh3nW .edge-pattern-dashed{stroke-dasharray:3;}#mermaid-svg-3IUxoHcixdkHh3nW .edge-pattern-dotted{stroke-dasharray:2;}#mermaid-svg-3IUxoHcixdkHh3nW .marker{fill:#333333;stroke:#333333;}#mermaid-svg-3IUxoHcixdkHh3nW .marker.cross{stroke:#333333;}#mermaid-svg-3IUxoHcixdkHh3nW svg{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;}#mermaid-svg-3IUxoHcixdkHh3nW p{margin:0;}#mermaid-svg-3IUxoHcixdkHh3nW .label{font-family:"trebuchet ms",verdana,arial,sans-serif;color:#333;}#mermaid-svg-3IUxoHcixdkHh3nW .cluster-label text{fill:#333;}#mermaid-svg-3IUxoHcixdkHh3nW .cluster-label span{color:#333;}#mermaid-svg-3IUxoHcixdkHh3nW .cluster-label span p{background-color:transparent;}#mermaid-svg-3IUxoHcixdkHh3nW .label text,#mermaid-svg-3IUxoHcixdkHh3nW span{fill:#333;color:#333;}#mermaid-svg-3IUxoHcixdkHh3nW .node rect,#mermaid-svg-3IUxoHcixdkHh3nW .node circle,#mermaid-svg-3IUxoHcixdkHh3nW .node ellipse,#mermaid-svg-3IUxoHcixdkHh3nW .node polygon,#mermaid-svg-3IUxoHcixdkHh3nW .node path{fill:#ECECFF;stroke:#9370DB;stroke-width:1px;}#mermaid-svg-3IUxoHcixdkHh3nW .rough-node .label text,#mermaid-svg-3IUxoHcixdkHh3nW .node .label text,#mermaid-svg-3IUxoHcixdkHh3nW .image-shape .label,#mermaid-svg-3IUxoHcixdkHh3nW .icon-shape .label{text-anchor:middle;}#mermaid-svg-3IUxoHcixdkHh3nW .node .katex path{fill:#000;stroke:#000;stroke-width:1px;}#mermaid-svg-3IUxoHcixdkHh3nW .rough-node .label,#mermaid-svg-3IUxoHcixdkHh3nW .node .label,#mermaid-svg-3IUxoHcixdkHh3nW .image-shape .label,#mermaid-svg-3IUxoHcixdkHh3nW .icon-shape .label{text-align:center;}#mermaid-svg-3IUxoHcixdkHh3nW .node.clickable{cursor:pointer;}#mermaid-svg-3IUxoHcixdkHh3nW .root .anchor path{fill:#333333!important;stroke-width:0;stroke:#333333;}#mermaid-svg-3IUxoHcixdkHh3nW .arrowheadPath{fill:#333333;}#mermaid-svg-3IUxoHcixdkHh3nW .edgePath .path{stroke:#333333;stroke-width:2.0px;}#mermaid-svg-3IUxoHcixdkHh3nW .flowchart-link{stroke:#333333;fill:none;}#mermaid-svg-3IUxoHcixdkHh3nW .edgeLabel{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-3IUxoHcixdkHh3nW .edgeLabel p{background-color:rgba(232,232,232, 0.8);}#mermaid-svg-3IUxoHcixdkHh3nW .edgeLabel rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-3IUxoHcixdkHh3nW .labelBkg{background-color:rgba(232, 232, 232, 0.5);}#mermaid-svg-3IUxoHcixdkHh3nW .cluster rect{fill:#ffffde;stroke:#aaaa33;stroke-width:1px;}#mermaid-svg-3IUxoHcixdkHh3nW .cluster text{fill:#333;}#mermaid-svg-3IUxoHcixdkHh3nW .cluster span{color:#333;}#mermaid-svg-3IUxoHcixdkHh3nW div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:12px;background:hsl(80, 100%, 96.2745098039%);border:1px solid #aaaa33;border-radius:2px;pointer-events:none;z-index:100;}#mermaid-svg-3IUxoHcixdkHh3nW .flowchartTitleText{text-anchor:middle;font-size:18px;fill:#333;}#mermaid-svg-3IUxoHcixdkHh3nW rect.text{fill:none;stroke-width:0;}#mermaid-svg-3IUxoHcixdkHh3nW .icon-shape,#mermaid-svg-3IUxoHcixdkHh3nW .image-shape{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-3IUxoHcixdkHh3nW .icon-shape p,#mermaid-svg-3IUxoHcixdkHh3nW .image-shape p{background-color:rgba(232,232,232, 0.8);padding:2px;}#mermaid-svg-3IUxoHcixdkHh3nW .icon-shape .label rect,#mermaid-svg-3IUxoHcixdkHh3nW .image-shape .label rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-3IUxoHcixdkHh3nW .label-icon{display:inline-block;height:1em;overflow:visible;vertical-align:-0.125em;}#mermaid-svg-3IUxoHcixdkHh3nW .node .label-icon path{fill:currentColor;stroke:revert;stroke-width:revert;}#mermaid-svg-3IUxoHcixdkHh3nW :root{--mermaid-font-family:"trebuchet ms",verdana,arial,sans-serif;} 节点 A

value:?

parent:null

left: B

right: C
节点 B

value:?

parent: A

left: null

right: null
节点 C

value:?

parent: A

left: null

right: null

A 的左子节点记录了 B 的地址，右子节点记录了 C 的地址；B 和 C 的父节点记录了 A 的地址。B 和 C 没有子节点，所以它们的左、右子节点均为 null。

2 ) 什么是二叉树

二叉树是指任意一个节点的子节点数量不超过 2 的树。子节点的数量在树结构中有一个专业术语叫做度（degree）。二叉树中每个节点的度 ≤ 2。

一个简单的二叉树示例：
#mermaid-svg-j8k88gLhOkv0RRbx{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;fill:#333;}@keyframes edge-animation-frame{from{stroke-dashoffset:0;}}@keyframes dash{to{stroke-dashoffset:0;}}#mermaid-svg-j8k88gLhOkv0RRbx .edge-animation-slow{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 50s linear infinite;stroke-linecap:round;}#mermaid-svg-j8k88gLhOkv0RRbx .edge-animation-fast{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 20s linear infinite;stroke-linecap:round;}#mermaid-svg-j8k88gLhOkv0RRbx .error-icon{fill:#552222;}#mermaid-svg-j8k88gLhOkv0RRbx .error-text{fill:#552222;stroke:#552222;}#mermaid-svg-j8k88gLhOkv0RRbx .edge-thickness-normal{stroke-width:1px;}#mermaid-svg-j8k88gLhOkv0RRbx .edge-thickness-thick{stroke-width:3.5px;}#mermaid-svg-j8k88gLhOkv0RRbx .edge-pattern-solid{stroke-dasharray:0;}#mermaid-svg-j8k88gLhOkv0RRbx .edge-thickness-invisible{stroke-width:0;fill:none;}#mermaid-svg-j8k88gLhOkv0RRbx .edge-pattern-dashed{stroke-dasharray:3;}#mermaid-svg-j8k88gLhOkv0RRbx .edge-pattern-dotted{stroke-dasharray:2;}#mermaid-svg-j8k88gLhOkv0RRbx .marker{fill:#333333;stroke:#333333;}#mermaid-svg-j8k88gLhOkv0RRbx .marker.cross{stroke:#333333;}#mermaid-svg-j8k88gLhOkv0RRbx svg{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;}#mermaid-svg-j8k88gLhOkv0RRbx p{margin:0;}#mermaid-svg-j8k88gLhOkv0RRbx .label{font-family:"trebuchet ms",verdana,arial,sans-serif;color:#333;}#mermaid-svg-j8k88gLhOkv0RRbx .cluster-label text{fill:#333;}#mermaid-svg-j8k88gLhOkv0RRbx .cluster-label span{color:#333;}#mermaid-svg-j8k88gLhOkv0RRbx .cluster-label span p{background-color:transparent;}#mermaid-svg-j8k88gLhOkv0RRbx .label text,#mermaid-svg-j8k88gLhOkv0RRbx span{fill:#333;color:#333;}#mermaid-svg-j8k88gLhOkv0RRbx .node rect,#mermaid-svg-j8k88gLhOkv0RRbx .node circle,#mermaid-svg-j8k88gLhOkv0RRbx .node ellipse,#mermaid-svg-j8k88gLhOkv0RRbx .node polygon,#mermaid-svg-j8k88gLhOkv0RRbx .node path{fill:#ECECFF;stroke:#9370DB;stroke-width:1px;}#mermaid-svg-j8k88gLhOkv0RRbx .rough-node .label text,#mermaid-svg-j8k88gLhOkv0RRbx .node .label text,#mermaid-svg-j8k88gLhOkv0RRbx .image-shape .label,#mermaid-svg-j8k88gLhOkv0RRbx .icon-shape .label{text-anchor:middle;}#mermaid-svg-j8k88gLhOkv0RRbx .node .katex path{fill:#000;stroke:#000;stroke-width:1px;}#mermaid-svg-j8k88gLhOkv0RRbx .rough-node .label,#mermaid-svg-j8k88gLhOkv0RRbx .node .label,#mermaid-svg-j8k88gLhOkv0RRbx .image-shape .label,#mermaid-svg-j8k88gLhOkv0RRbx .icon-shape .label{text-align:center;}#mermaid-svg-j8k88gLhOkv0RRbx .node.clickable{cursor:pointer;}#mermaid-svg-j8k88gLhOkv0RRbx .root .anchor path{fill:#333333!important;stroke-width:0;stroke:#333333;}#mermaid-svg-j8k88gLhOkv0RRbx .arrowheadPath{fill:#333333;}#mermaid-svg-j8k88gLhOkv0RRbx .edgePath .path{stroke:#333333;stroke-width:2.0px;}#mermaid-svg-j8k88gLhOkv0RRbx .flowchart-link{stroke:#333333;fill:none;}#mermaid-svg-j8k88gLhOkv0RRbx .edgeLabel{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-j8k88gLhOkv0RRbx .edgeLabel p{background-color:rgba(232,232,232, 0.8);}#mermaid-svg-j8k88gLhOkv0RRbx .edgeLabel rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-j8k88gLhOkv0RRbx .labelBkg{background-color:rgba(232, 232, 232, 0.5);}#mermaid-svg-j8k88gLhOkv0RRbx .cluster rect{fill:#ffffde;stroke:#aaaa33;stroke-width:1px;}#mermaid-svg-j8k88gLhOkv0RRbx .cluster text{fill:#333;}#mermaid-svg-j8k88gLhOkv0RRbx .cluster span{color:#333;}#mermaid-svg-j8k88gLhOkv0RRbx div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:12px;background:hsl(80, 100%, 96.2745098039%);border:1px solid #aaaa33;border-radius:2px;pointer-events:none;z-index:100;}#mermaid-svg-j8k88gLhOkv0RRbx .flowchartTitleText{text-anchor:middle;font-size:18px;fill:#333;}#mermaid-svg-j8k88gLhOkv0RRbx rect.text{fill:none;stroke-width:0;}#mermaid-svg-j8k88gLhOkv0RRbx .icon-shape,#mermaid-svg-j8k88gLhOkv0RRbx .image-shape{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-j8k88gLhOkv0RRbx .icon-shape p,#mermaid-svg-j8k88gLhOkv0RRbx .image-shape p{background-color:rgba(232,232,232, 0.8);padding:2px;}#mermaid-svg-j8k88gLhOkv0RRbx .icon-shape .label rect,#mermaid-svg-j8k88gLhOkv0RRbx .image-shape .label rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-j8k88gLhOkv0RRbx .label-icon{display:inline-block;height:1em;overflow:visible;vertical-align:-0.125em;}#mermaid-svg-j8k88gLhOkv0RRbx .node .label-icon path{fill:currentColor;stroke:revert;stroke-width:revert;}#mermaid-svg-j8k88gLhOkv0RRbx :root{--mermaid-font-family:"trebuchet ms",verdana,arial,sans-serif;} 2
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3 ) 树的基本术语

根节点（Root Node） ：树的最顶层节点，没有父节点。

子节点（Child Node） ：某个节点的直接后代。

父节点（Parent Node） ：某个节点的直接前驱。

叶子节点（Leaf Node） ：度为零的节点，没有子节点。

树的高度（Height） ：树中节点的最大层数。

左子树 / 右子树：以根节点的左子节点为根的子树称为左子树，以右子节点为根的称为右子树。

以高度为 4 的二叉树为例：
#mermaid-svg-M8TALdpAxiLdqAbX{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;fill:#333;}@keyframes edge-animation-frame{from{stroke-dashoffset:0;}}@keyframes dash{to{stroke-dashoffset:0;}}#mermaid-svg-M8TALdpAxiLdqAbX .edge-animation-slow{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 50s linear infinite;stroke-linecap:round;}#mermaid-svg-M8TALdpAxiLdqAbX .edge-animation-fast{stroke-dasharray:9,5!important;stroke-dashoffset:900;animation:dash 20s linear infinite;stroke-linecap:round;}#mermaid-svg-M8TALdpAxiLdqAbX .error-icon{fill:#552222;}#mermaid-svg-M8TALdpAxiLdqAbX .error-text{fill:#552222;stroke:#552222;}#mermaid-svg-M8TALdpAxiLdqAbX .edge-thickness-normal{stroke-width:1px;}#mermaid-svg-M8TALdpAxiLdqAbX .edge-thickness-thick{stroke-width:3.5px;}#mermaid-svg-M8TALdpAxiLdqAbX .edge-pattern-solid{stroke-dasharray:0;}#mermaid-svg-M8TALdpAxiLdqAbX .edge-thickness-invisible{stroke-width:0;fill:none;}#mermaid-svg-M8TALdpAxiLdqAbX .edge-pattern-dashed{stroke-dasharray:3;}#mermaid-svg-M8TALdpAxiLdqAbX .edge-pattern-dotted{stroke-dasharray:2;}#mermaid-svg-M8TALdpAxiLdqAbX .marker{fill:#333333;stroke:#333333;}#mermaid-svg-M8TALdpAxiLdqAbX .marker.cross{stroke:#333333;}#mermaid-svg-M8TALdpAxiLdqAbX svg{font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:16px;}#mermaid-svg-M8TALdpAxiLdqAbX p{margin:0;}#mermaid-svg-M8TALdpAxiLdqAbX .label{font-family:"trebuchet ms",verdana,arial,sans-serif;color:#333;}#mermaid-svg-M8TALdpAxiLdqAbX .cluster-label text{fill:#333;}#mermaid-svg-M8TALdpAxiLdqAbX .cluster-label span{color:#333;}#mermaid-svg-M8TALdpAxiLdqAbX .cluster-label span p{background-color:transparent;}#mermaid-svg-M8TALdpAxiLdqAbX .label text,#mermaid-svg-M8TALdpAxiLdqAbX span{fill:#333;color:#333;}#mermaid-svg-M8TALdpAxiLdqAbX .node rect,#mermaid-svg-M8TALdpAxiLdqAbX .node circle,#mermaid-svg-M8TALdpAxiLdqAbX .node ellipse,#mermaid-svg-M8TALdpAxiLdqAbX .node polygon,#mermaid-svg-M8TALdpAxiLdqAbX .node path{fill:#ECECFF;stroke:#9370DB;stroke-width:1px;}#mermaid-svg-M8TALdpAxiLdqAbX .rough-node .label text,#mermaid-svg-M8TALdpAxiLdqAbX .node .label text,#mermaid-svg-M8TALdpAxiLdqAbX .image-shape .label,#mermaid-svg-M8TALdpAxiLdqAbX .icon-shape .label{text-anchor:middle;}#mermaid-svg-M8TALdpAxiLdqAbX .node .katex path{fill:#000;stroke:#000;stroke-width:1px;}#mermaid-svg-M8TALdpAxiLdqAbX .rough-node .label,#mermaid-svg-M8TALdpAxiLdqAbX .node .label,#mermaid-svg-M8TALdpAxiLdqAbX .image-shape .label,#mermaid-svg-M8TALdpAxiLdqAbX .icon-shape .label{text-align:center;}#mermaid-svg-M8TALdpAxiLdqAbX .node.clickable{cursor:pointer;}#mermaid-svg-M8TALdpAxiLdqAbX .root .anchor path{fill:#333333!important;stroke-width:0;stroke:#333333;}#mermaid-svg-M8TALdpAxiLdqAbX .arrowheadPath{fill:#333333;}#mermaid-svg-M8TALdpAxiLdqAbX .edgePath .path{stroke:#333333;stroke-width:2.0px;}#mermaid-svg-M8TALdpAxiLdqAbX .flowchart-link{stroke:#333333;fill:none;}#mermaid-svg-M8TALdpAxiLdqAbX .edgeLabel{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-M8TALdpAxiLdqAbX .edgeLabel p{background-color:rgba(232,232,232, 0.8);}#mermaid-svg-M8TALdpAxiLdqAbX .edgeLabel rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-M8TALdpAxiLdqAbX .labelBkg{background-color:rgba(232, 232, 232, 0.5);}#mermaid-svg-M8TALdpAxiLdqAbX .cluster rect{fill:#ffffde;stroke:#aaaa33;stroke-width:1px;}#mermaid-svg-M8TALdpAxiLdqAbX .cluster text{fill:#333;}#mermaid-svg-M8TALdpAxiLdqAbX .cluster span{color:#333;}#mermaid-svg-M8TALdpAxiLdqAbX div.mermaidTooltip{position:absolute;text-align:center;max-width:200px;padding:2px;font-family:"trebuchet ms",verdana,arial,sans-serif;font-size:12px;background:hsl(80, 100%, 96.2745098039%);border:1px solid #aaaa33;border-radius:2px;pointer-events:none;z-index:100;}#mermaid-svg-M8TALdpAxiLdqAbX .flowchartTitleText{text-anchor:middle;font-size:18px;fill:#333;}#mermaid-svg-M8TALdpAxiLdqAbX rect.text{fill:none;stroke-width:0;}#mermaid-svg-M8TALdpAxiLdqAbX .icon-shape,#mermaid-svg-M8TALdpAxiLdqAbX .image-shape{background-color:rgba(232,232,232, 0.8);text-align:center;}#mermaid-svg-M8TALdpAxiLdqAbX .icon-shape p,#mermaid-svg-M8TALdpAxiLdqAbX .image-shape p{background-color:rgba(232,232,232, 0.8);padding:2px;}#mermaid-svg-M8TALdpAxiLdqAbX .icon-shape .label rect,#mermaid-svg-M8TALdpAxiLdqAbX .image-shape .label rect{opacity:0.5;background-color:rgba(232,232,232, 0.8);fill:rgba(232,232,232, 0.8);}#mermaid-svg-M8TALdpAxiLdqAbX .label-icon{display:inline-block;height:1em;overflow:visible;vertical-align:-0.125em;}#mermaid-svg-M8TALdpAxiLdqAbX .node .label-icon path{fill:currentColor;stroke:revert;stroke-width:revert;}#mermaid-svg-M8TALdpAxiLdqAbX :root{--mermaid-font-family:"trebuchet ms",verdana,arial,sans-serif;} 2
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  • 第 1 层：节点 7
  • 第 2 层：节点 4、10
  • 第 3 层：节点 2、5、11
  • 第 4 层：节点 12

树的高度 = 层数 = 4。

  • 7 是根节点。
  • 4 是 7 的左子节点，10 是 7 的右子节点。
  • 蓝色虚线框住的部分（以 4 为根的子树）是根节点的左子树，高度为 3。
  • 绿色虚线框住的部分（以 10 为根的子树）是根节点的右子树，高度为 3。

二叉查找树（二叉排序树）

1 ) 定义与特点

普通的二叉树在存储数据时没有任何规律，查找某个节点需要遍历整棵树，效率很低。

为了提高查找效率，引入了二叉查找树（Binary Search Tree，BST），也称为二叉排序树或二叉搜索树。

二叉查找树必须满足：

  1. 首先是一棵二叉树（每个节点最多两个子节点）。
  2. 对于任意节点，其左子节点的值 < 该节点的值。
  3. 对于任意节点，其右子节点的值 > 该节点的值。

简单理解：任意节点从左到右依次增大。

示例：
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4
5
7
9
10
11

  • 7 的左子节点 4 < 7，右子节点 10 > 7。
  • 4 的左子节点 2 < 4，右子节点 5 > 4。
  • 10 的左子节点 9 < 10，右子节点 11 > 10。

这是一棵标准的二叉查找树。

2 ) 添加节点过程

向二叉查找树中添加节点时，遵循小的存左边，大的存右边，相等的值不存 的规则。

具体过程如下：

  1. 如果树为空，直接将新节点作为根节点。
  2. 如果树不为空，从根节点开始比较：
    • 若新节点的值 < 当前节点的值，且当前节点的左子节点为空，则成为左子节点；否则进入左子树继续比较。
    • 若新节点的值 > 当前节点的值，且当前节点的右子节点为空，则成为右子节点；否则进入右子树继续比较。
    • 若新节点的值 == 当前节点的值，不存储（或根据具体需求处理）。

添加实例：依次添加 7、4、10、5。

  1. 添加 7：树空，7 成为根节点。
  2. 添加 4：4 < 7，且 7 的左子节点为空，所以 4 成为 7 的左子节点。
  3. 添加 10：10 > 7，且 7 的右子节点为空，所以 10 成为 7 的右子节点。
  4. 添加 5：5 < 7，向左走；5 > 4，且 4 的右子节点为空，所以 5 成为 4 的右子节点。

结果：
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5
7
10

如果继续添加 12、13，会出现"一边倒"的情况，这在后面会引出平衡二叉树的需求。

3 ) 代码演示：节点类与插入

下面使用 Java 代码演示二叉查找树的节点结构和插入过程。

节点类（包含父节点引用，以备平衡树使用）：

java 复制代码 
class TreeNode {
    int value;
    TreeNode parent;
    TreeNode left;
    TreeNode right;
 
    public TreeNode(int value) {
        this.value = value;
    }
 
    // 插入节点（二叉查找树规则）
    public void insert(int newValue) {
        if (newValue < this.value) {
            if (left == null) {
                left = new TreeNode(newValue);
                left.parent = this;
            } else {
                left.insert(newValue);
            }
        } else if (newValue > this.value) {
            if (right == null) {
                right = new TreeNode(newValue);
                right.parent = this;
            } else {
                right.insert(newValue);
            }
        }
        // 相等则不处理 
    }
}

构建树并输出结构：

java 复制代码 
public class BSTDemo {
    public static void main(String[] args) {
        TreeNode root = new TreeNode(7);
        root.insert(4);
        root.insert(10);
        root.insert(5);
        // 输出树的结构（先序）
        printTree(root, "");
    }
 
    public static void printTree(TreeNode node, String prefix) {
        if (node == null) return;
        System.out.println(prefix + "├─ " + node.value);
        if (node.left != null || node.right != null) {
            if (node.left != null) {
                printTree(node.left, prefix + "│   ");
            } else {
                System.out.println(prefix + "│   ├─ null");
            }
            if (node.right != null) {
                printTree(node.right, prefix + "    ");
            } else {
                System.out.println(prefix + "    ├─ null");
            }
        }
    }
}

输出示例：

tree 复制代码 
├─ 7 
│   ├─ 4 
│   │   ├─ null 
│   │   ├─ 5 
│   │       ├─ null 
│   │       ├─ null 
│   ├─ 10 
│       ├─ null 
│       ├─ null

平衡二叉树

1 ) 为什么需要平衡

二叉查找树的插入顺序会影响树的形状。例如按 7、10、11、12、13 的顺序依次插入：
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10
11
12
13

整棵树退化成了一条链表。此时查找节点 13 需要比较 5 次，效率降低。树的高度差越大，查询效率就越接近 O(n)。为了避免这种情况，需要让树的左右子树高度尽可能接近，即平衡。

2 ) 定义与判断

平衡二叉树（AVL 树是最典型的一种）的定义：

  1. 二叉树中，左右子树的高度差不超过 1。
  2. 任意节点的左右子树也是平衡二叉树。

判断示例：

二叉树 是否平衡 原因
文本描述：根节点 7，左子树高度 3，右子树高度 4，但节点 10 的左子树高度 0，右子树高度 3，差超过 1 否 节点 10 左右高度差为 3 − 0 = 3 > 1
文本描述：节点 4 左子树高度 2，右子树高度 0 否 节点 4 左右高度差 2 − 0 = 2 > 1
文本描述：根节点 7，左子树高度 2，右子树高度 1；其他节点高度差均 ≤ 1 是 所有节点高度差都 ≤ 1
文本描述：根节点 7 左右高度均 2；节点 4 左高度 1，右高度 0，差 1 是 所有节点高度差 ≤ 1

平衡二叉树的查询效率始终保持在 O(log n) 级别，因为树的高度被控制在对数级别。

平衡二叉树的旋转机制

当向平衡二叉树中添加一个节点后，可能导致树不再平衡。此时，需要通过左旋 和右旋操作来恢复平衡。

1 ) 左旋（Left Rotation）

触发条件：右子树过高（右侧节点偏多），需要将右侧节点向左旋转。

左旋步骤：

  1. 将原先的右子节点（设为 R）提升为新的根节点。
  2. 原先的根节点（设为 E）降级为 R 的左子节点。
  3. R 原先的左子节点移交给 E，作为 E 的右子节点。

简单左旋图解：

旋转前：
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左子节点
R
RL
RR

旋转后：
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E 左子节点
RR 右子节点
原左子节点
RL 原 R 的左子节点

复杂左旋图解（R 节点有一个左子节点）：

旋转前：
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  • 添加节点 12 后，右子树过高，触发左旋。
  • 先忽略 9（10 的左子节点），将 10 提升为新根，7 降级为 10 的左子节点。
  • 处理 9：原来 9 是 10 的左子节点，旋转后无法再作为 10 的左子节点（因为 7 已经是左子节点了），因此将 9 移交给降级后的 7，作为 7 的右子节点。

旋转后：
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Java 代码演示左旋（假设节点包含父指针）：

java 复制代码 
// 左旋操作，以当前节点为根进行左旋 
public TreeNode rotateLeft() {
    TreeNode newRoot = this.right;  // 新根是当前的右子节点 
    if (newRoot == null) return this;
 
    // 当前节点成为新根的左子节点 
    this.right = newRoot.left;
    if (newRoot.left != null) {
        newRoot.left.parent = this;
    }
 
    // 新根替换当前节点的位置 
    newRoot.parent = this.parent;
    if (this.parent != null) {
        if (this.parent.left == this) {
            this.parent.left = newRoot;
        } else {
            this.parent.right = newRoot;
        }
    }
 
    // 设置当前节点为新根的左子节点 
    newRoot.left = this;
    this.parent = newRoot;
 
    return newRoot; // 返回新的根节点 
}

2 ) 右旋（Right Rotation）

触发条件：左子树过高，需要将左侧节点向右旋转。

右旋步骤：

  1. 将原先的左子节点（设为 L）提升为新的根节点。
  2. 原先的根节点（设为 E）降级为 L 的右子节点。
  3. L 原先的右子节点移交给 E，作为 E 的左子节点。

简单右旋图解：

旋转前：
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L
右子节点
LL
LR

旋转后：
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LL
E 右子节点
LR
右子节点

复杂右旋图解（L 节点有一个右子节点 5）：

旋转前：
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  • 添加节点 1 后，左子树过高，触发右旋。
  • 先忽略 5（4 的右子节点），将 4 提升为新根，7 降级为 4 的右子节点。
  • 处理 5：原来 5 是 4 的右子节点，旋转后交给降级后的 7，作为 7 的左子节点。

旋转后：
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Java 代码演示右旋：

java 复制代码 
// 右旋操作，以当前节点为根进行右旋 
public TreeNode rotateRight() {
    TreeNode newRoot = this.left;   // 新根是当前的左子节点 
    if (newRoot == null) return this;
 
    // 当前节点成为新根的右子节点 
    this.left = newRoot.right;
    if (newRoot.right != null) {
        newRoot.right.parent = this;
    }
 
    // 新根替换当前节点的位置 
    newRoot.parent = this.parent;
    if (this.parent != null) {
        if (this.parent.left == this) {
            this.parent.left = newRoot;
        } else {
            this.parent.right = newRoot;
        }
    }
 
    // 设置当前节点为新根的右子节点 
    newRoot.right = this;
    this.parent = newRoot;
 
    return newRoot; // 返回新的根节点 
}

3 ) 左左、左右、右右、右左四种情况

当插入节点导致失衡时，根据插入位置的不同，分为四种情况，分别采用不同的旋转组合来恢复平衡。

情况 插入位置 失衡特征 恢复操作
左左 (LL) 根节点左子树的左子树 左子树高度 − 右子树高度 > 1，且左子节点的左子树高 一次右旋
左右 (LR) 根节点左子树的右子树 左子树高度 − 右子树高度 > 1，且左子节点的右子树高 先局部左旋，再整体右旋
右右 (RR) 根节点右子树的右子树 右子树高度 − 左子树高度 > 1，且右子节点的右子树高 一次左旋
右左 (RL) 根节点右子树的左子树 右子树高度 − 左子树高度 > 1，且右子节点的左子树高 先局部右旋，再整体左旋

左左 (LL) 示例：
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插入后失衡
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一次右旋后平衡

左右 (LR) 示例（在左子树的右子树插入 6）：

失衡树：
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  • 第一步：对 4 的右子树进行左旋（局部），5 上提，4 下移。
  • 第二步：得到类似"左左"的结构后，再对整棵树进行右旋。

旋转过程：
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局部左旋结果
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第一步局部左旋
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第一步局部左旋
局部左旋结果
第二步整体右旋结果

右右 (RR)、右左 (RL) 与左左、左右镜像对称，不再赘述。

判断与恢复平衡的伪代码：

java 复制代码 
// 插入节点后，从插入点向上回溯，检查每个节点的平衡因子 
void rebalance(TreeNode node) {
    while (node != null) {
        // 计算平衡因子 = 左子树高度 - 右子树高度 
        int balance = getHeight(node.left) - getHeight(node.right);
 
        if (balance > 1) {                 // 左子树过高 
            if (getHeight(node.left.left) >= getHeight(node.left.right)) {
                // 左左情况 → 一次右旋 
                node = node.rotateRight();
            } else {
                // 左右情况 → 先左旋左子节点，再右旋当前节点 
                node.left = node.left.rotateLeft();
                node = node.rotateRight();
            }
        } else if (balance < -1) {         // 右子树过高 
            if (getHeight(node.right.right) >= getHeight(node.right.left)) {
                // 右右情况 → 一次左旋 
                node = node.rotateLeft();
            } else {
                // 右左情况 → 先右旋右子节点，再左旋当前节点 
                node.right = node.right.rotateRight();
                node = node.rotateLeft();
            }
        } else {
            // 平衡，无需旋转 
        }
        // 向上回溯 
        node = node.parent;
    }
}

总结

本文从最基本的节点结构开始，循序渐进地介绍了二叉树、二叉查找树和平衡二叉树的概念与操作，并重点讲解了平衡二叉树通过左旋、右旋维持平衡的机制。

理解这些数据结构是学习 Java 中 TreeSet、TreeMap 底层红黑树的基础。红黑树是一种更复杂的自平衡二叉查找树，它同样依赖旋转操作来保持近似平衡，从而保证查找、插入、删除的最坏时间复杂度为 O(log n)。

联系红黑树的结构和规则，进一步完善对 TreeSet 底层实现的理解。

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