c#实现b树

using System;
using System.Collections.Generic;
using System.Linq;

namespace BTree
{
    class Program
    {
        static void Main(string[] args)
        {
            #region B树的定义

            //1.有x.n个关键字,x.leaf是否为叶子节点(是叶子节点为true),x.key1<=x.key2<=...x.keyn
            //2.有x.n+1个x.ci指向子的指针
            //3.父的关键字ki大于之前指针指向子树的所有关键字.小于之后的子树的所有关键字
            //4.所有叶节点高度一样
            //5.节点的关键字数目限定和子树数目的限定. 对于t>=2. 关键字的范围为 t-1~2t-1(根除外,可以只有一个关键字) ,子树范围为t~2t
            //其实5的定义和阶数定义一样.m阶的b树.且根最少两个子树. 非叶子节点至少floor(m/2)-1个关键字,至多m个.则2t=m

            BTree<int> tree=new BTree<int>(3);
            //测试插入
            BTreeInsertNode(tree,15);
            BTreeInsertNode(tree, 16);
            BTreeInsertNode(tree, 25);
            BTreeInsertNode(tree, 34);
            BTreeInsertNode(tree, 58);
            BTreeInsertNode(tree, 19); 
            BTreeInsertNode(tree, 68); 
            BTreeInsertNode(tree, 78); 
            BTreeInsertNode(tree, 95);
            //测试查找
            Console.WriteLine(BTreeSearch(tree.Root,78).Position);
            //测试删除
            BTreeDeleteNode(tree,15);
            Console.WriteLine(BTreeSearch(tree.Root,15)==null);

            #endregion

        }

        #region 树节点数据结构定义

        /// <summary>
        /// B树节点的定义
        /// </summary>
        /// <typeparam name="T"></typeparam>
        public class BTreeNode<T> where T: IComparable<T>
        {
            /// <summary>
            /// Key关键字数量(n)
            /// </summary>
            public int KeyCount { get; set; }

            /// <summary>
            /// 关键字数组(n)
            /// </summary>
            public T[] Keys { get; set; }

            /// <summary>
            /// 孩子指针(n+1)
            /// </summary>
            public BTreeNode<T>[] Childs { get; set; }

            /// <summary>
            /// 是否叶子节点(是则为true)
            /// </summary>
            public bool IsLeaf { get; set; }

            /// <summary>
            /// 节点阶数,因为节点的数组大小受到是m阶b树的限制
            /// </summary>
            public int Order { get; }

            /// <summary>
            /// 初始化节点,数组为最大阶数大小
            /// n==keycount必有n个关键字,必有n+1个孩子指针,为叶节点时孩子全部为null
            /// </summary>
            /// <param name="order">阶数</param>
            public BTreeNode(int order)
            {
                KeyCount = 0;
                Keys=new T[order];
                Childs=new BTreeNode<T>[order+1];
                IsLeaf = true;
                Order = order;
            }

            /// <summary>
            /// 清理大于KeyCount的数组位置为默认值
            /// </summary>
            public void Clear()
            {
                if (this.Keys != null)
                {
                    for (int i = KeyCount; i < Order; i++)
                    {
                        this.Keys[i] = default(T);
                    }
                }
            }
        }

        #endregion

        #region B树的定义

        public class BTree<T> where T:IComparable<T>
        {
            /// <summary>
            /// 根节点
            /// </summary>
            public BTreeNode<T> Root { get; set; }

            /// <summary>
            /// 阶数
            /// </summary>
            public int Order { get; }

            /// <summary>
            /// 构造时指定阶数对树进行限定
            /// </summary>
            /// <param name="order"></param>
            public BTree(int order)
            {
                Order = order;
                Root=new BTreeNode<T>(order);
            }

            /// <summary>
            /// 树节点是否能被添加
            /// </summary>
            /// <returns></returns>
            public bool IsCanAdded(BTreeNode<T> node)
            {
                return node.Order == Order;
            }
        }

        #endregion

        #region 创建B树

        public static BTree<int> CreateBTree(int order)
        {
            if(order%2!=1) throw new Exception("传入的阶数不合法!");
            BTree<int> tree=new BTree<int>(order);
            return tree;
        }

        #endregion

        #region B树查找返回值

        public class BTreeSearchReturnValue<T> where T:IComparable<T>
        {
            public BTreeNode<T> Node { get; set; }

            public int Position { get; set; }

            public BTreeSearchReturnValue(BTreeNode<T> node, int position)
            {
                Node = node;
                Position = position;
            }
        }

        #endregion

        #region 节点查找

        public static BTreeSearchReturnValue<int> BTreeSearch(BTreeNode<int> x,int k)
        {
            int i = 0;
            while (i<x.KeyCount && x.Keys[i]<k)
            {
                i++;
            }

            if(i<x.KeyCount && x.Keys[i]==k) return new BTreeSearchReturnValue<int>(x,i);
            else if (x.IsLeaf) return null;
            else return BTreeSearch(x.Childs[i], k); 
        }

        #endregion

        #region 节点的分裂

        public static void BTreeSplitNode(BTreeNode<int> x,int i)
        {
            int splitLength = (x.Order + 1) / 2 - 1;
            BTreeNode<int> z=new BTreeNode<int>(x.Order);
            BTreeNode<int> y = x.Childs[i];
            z.IsLeaf = y.IsLeaf;
            z.KeyCount = splitLength;
            for (int j = 0; j < splitLength; j++)
            {
                z.Keys[j] = y.Keys[splitLength + j+1];
            }

            if (!y.IsLeaf)
            {
                for (int j = 0; j <= splitLength; j++)
                {
                    z.Childs[j] = y.Childs[splitLength +1+ j];
                }
            }

            y.KeyCount = splitLength;
            for (int j = x.KeyCount; j >=i+1; j--)
            {
                x.Childs[j + 1] = x.Childs[j];
            }

            x.Childs[i + 1] = z;
            for (int j = x.KeyCount-1; j >=i; j--)
            {
                x.Keys[j + 1] = x.Keys[j];
            }

            x.Keys[i] = y.Keys[splitLength];
            x.KeyCount++;
            y.Clear();//清理大于keycount位置的元素,也可以不清理
        }

        #endregion

        #region 节点的插入(非满)

        public static void BTreeInsertNodeNotFull(BTreeNode<int> x,int k)
        {
            int i = x.KeyCount-1;
            if (x.IsLeaf)
            {
                while (i>=0 && x.Keys[i]>k)
                {
                    x.Keys[i + 1] = x.Keys[i];
                    i--;
                }

                x.Keys[i + 1] = k;
                x.KeyCount++;
            }
            else
            {
                while (i >= 0 && x.Keys[i] > k)
                {
                    i--;
                }

                i = i + 1;
                if (x.Childs[i].KeyCount == x.Order)
                {
                    BTreeSplitNode(x,i);
                    if (x.Keys[i] < k) i++;
                }
                BTreeInsertNodeNotFull(x.Childs[i],k);
            }
        }

        #endregion

        #region 节点插入(从根单程向下)

        public static void BTreeInsertNode(BTree<int> t,int k)
        {
            BTreeNode<int> root = t.Root;
            if (root.KeyCount == root.Order)
            {
                BTreeNode<int> s=new BTreeNode<int>(root.Order);
                t.Root = s;
                s.IsLeaf = false;
                s.Childs[0] = root;
                BTreeSplitNode(s,0);
                BTreeInsertNodeNotFull(s,k);
            }
            else BTreeInsertNodeNotFull(root, k);
        }

        #endregion

        #region 前驱(只找叶节点方向)

        public static int BTreeFindPreDecessor(BTreeNode<int> x)
        {
            BTreeNode<int> y = x;
            while (!y.IsLeaf)
            {
                y = y.Childs[y.KeyCount];
            }

            return y.Keys[y.KeyCount-1];
        }

        #endregion

        #region 后继(只找叶节点方向)

        public static int BTreeSearchSuccessor(BTreeNode<int> x)
        {
            BTreeNode<int> y = x;
            while (!y.IsLeaf)
            {
                y = y.Childs[0];
            }

            return y.Keys[0];
        }

        #endregion

        #region B树的节点删除

        //1.如果关键字k在节点x中,并且x是叶子节点,则从x中删除k;
        //2.如果关键字在结点x中,并且x是内部节点,则做以下操作
                //a.如果x中前于k的子节点y至少包含t个关键字,则找出k在以y为根的子树中的前驱k',递归的删除k',并在x中用k'代替k.
                //b.对称的,如果y少于t个关键字,则检查x后于k的节点z,若z至少有t个关键字.则找出z的后继k'.递归的删除k'.并在x中使用k'代替k
                //c.否则:y和z都只包含t-1个关键字.则将k和z全部合并放入y(2t-1),这样x就失去了k和指向z的指针,释放z,并且在y中递归的删除k
        //3.如果关键字k不在当前节点x中,则必在包含k的子树x.Ci中,如果x.Ci只有t-1个关键字,则执行3a或者3b保证节点至少含有t个节点.然后通过对x的某个合适的
        //子节点进行递归而结束
                //a.如果x.Ci只包含t-1个关键字,但是他的一个相邻兄弟至少包含t个关键字,则将x中的某一个关键字下降至x.Ci,将x.Ci相邻兄弟的一个关键字提升
                //至x,并且将该兄弟的孩子指针移动到x.ci中,这样就使得x.ci增加了一个额外的关键字.
                //b.如果x.ci以及相邻兄弟都只有t-1个关键字,则将x.ci与一个相邻兄弟合并,即将x的一个关键字移动至新合并的节点,使之成为新节点的中间关键字

        /// <summary>
        /// 特殊处理根节点,若根节点keycount==1且左右子树的keycount==t-1,则合并后删除
        /// </summary>
        /// <param name="t"></param>
        /// <param name="k"></param>
        public static void BTreeDeleteNode(BTree<int> t,int k)
        {
            BTreeNode<int> x = t.Root;
            if (x.KeyCount == 1)
            {
                BTreeNode<int> y = x.Childs[0];
                BTreeNode<int> z = x.Childs[1];
                if (y.KeyCount == (x.Order + 1) / 2 && z.KeyCount == (x.Order + 1) / 2)
                {
                    BTreeNodeMerge(x, 0, y, z);
                    t.Root = y;
                    BTreeDeleteNoNode(y, k);
                }
                else BTreeDeleteNoNode(x, k);
            }
            else BTreeDeleteNoNode(x,k);
        }

        public static void BTreeDeleteNoNode(BTreeNode<int> x,int k)
        {
            int i = 0;
            int t = (x.Order + 1) / 2;
            //获取删除位置,为i的key或者为i的子树上
            while (i < x.KeyCount && x.Keys[i] < k)
            {
                i++;
            }
            if (x.IsLeaf)
            {
                //1.如果关键字k在节点x中,并且x是叶子节点,则从x中删除k;
                if (x.Keys[i] == k)
                {
                    for (int j = i; j < x.KeyCount; j++)
                    {
                        x.Keys[j] = x.Keys[j+1];
                    }

                    x.KeyCount--;
                }
                else throw new Exception("不存在该关键字!");
            }
            else
            {
                BTreeNode<int> y = x.Childs[i];
                BTreeNode<int> z=null;
                if (i < x.KeyCount)
                {
                    z = x.Childs[i + 1];
                }

                if (i < x.KeyCount && x.Keys[i] == k)
                {
                    //2.如果关键字在结点x中,并且x是内部节点
                    if (y.KeyCount > t - 1)
                    {
                        //2.a
                        int newK = BTreeFindPreDecessor(y);
                        BTreeDeleteNoNode(y,newK);
                        x.Keys[i] = newK;
                    }
                    else if (z.KeyCount > t - 1)
                    {
                        //2.b
                        int newK = BTreeSearchSuccessor(z);
                        BTreeDeleteNoNode(z,newK);
                        x.Keys[i] = newK;
                    }
                    else
                    {
                        //2.c
                        BTreeNodeMerge(x,i,y,z);
                        BTreeDeleteNoNode(y,k);
                    }
                }
                else
                {
                    //3.如果关键字k不在当前节点x中,则必在包含k的子树x.Ci中,如果x.Ci只有t-1个关键字,则执行3a或者3b保证节点至少含有t个节点.然后通过对x的某个合适的
                    //子节点进行递归而结束
                    BTreeNode<int> p = null;
                    if (i > 0)
                    {
                        p = x.Childs[i-1];
                    }
                    if (y.KeyCount == t - 1)
                    {
                        if (i > 0 && p.KeyCount > t - 1)
                        {
                            //3.a-left
                            BTreeShiftToRightChild(x,i-1,p,y);
                        }
                        else if (i < x.KeyCount && z.KeyCount > t - 1)
                        {
                            //3.a-right
                            BTreeShifToLeftChild(x,i,y,z);
                        }
                        else if (i>0)
                        {
                            //3.b-merge left
                            BTreeNodeMerge(x,i-1,p,y);
                            y = p;
                        }
                        else BTreeNodeMerge(x,i,y,z);//3.b-merge right
                    }
                    BTreeDeleteNoNode(y,k);
                }
            }
        }

        #endregion

        #region 节点的合并

        /// <summary>
        /// 当左右两个节点都为t-1个关键字时,将x.keyi与z合并到y
        /// </summary>
        /// <param name="x">父节点(必不为t-1个关键字,若为则需要执行3.a,b使之满足)</param>
        /// <param name="i">要合并的子树索引</param>
        /// <param name="y">x的第i子树</param>
        /// <param name="z">x的第i+1子树</param>
        public static void BTreeNodeMerge(BTreeNode<int> x,int i,BTreeNode<int> y,BTreeNode<int> z)
        {
            int t = (y.Order + 1) / 2;
            y.KeyCount = 2*t-1;
            for (int j = t; j < y.KeyCount; j++)
            {
                y.Keys[j] = z.Keys[j-t];
            }

            y.Keys[t - 1] = x.Keys[i];
            if (!y.IsLeaf)
            {
                for (int j = t; j < y.KeyCount+1; j++)
                {
                    y.Childs[j] = x.Childs[j-t];
                }
            }

            for (int j = i; j < x.KeyCount-1; j++)
            {
                x.Keys[j] = x.Keys[j + 1];
            }

            for (int j = i+1; j < x.KeyCount; j++)
            {
                x.Childs[j] = x.Childs[j+1];
            }

            x.KeyCount--;
        }

        #endregion

        #region 从左边借节点,右边是带删除关键字的节点

        /// <summary>
        /// z为带删除关键字的节点,只有t-1个关键字.从左边大于t-1个关键字的节点借一个关键字
        /// </summary>
        /// <param name="x"></param>
        /// <param name="i"></param>
        /// <param name="y"></param>
        /// <param name="z"></param>
        public static void BTreeShiftToRightChild(BTreeNode<int> x,int i,BTreeNode<int> y,BTreeNode<int> z)
        {
            z.KeyCount++;
            for (int j = z.KeyCount; j >0; j++)
            {
                z.Keys[j] = z.Keys[j-1];
            }

            z.Keys[0] = x.Keys[i];
            x.Keys[i] = y.Keys[y.KeyCount-1];
            if (!z.IsLeaf)
            {
                for (int j = z.KeyCount+1; j >0; j++)
                {
                    z.Childs[j] = z.Childs[j-1];
                }

                z.Childs[0] = y.Childs[y.KeyCount];
            }

            y.KeyCount--;
        }

        #endregion

        #region 从右边借关键字,左边是带删除关键字的节点

        public static void BTreeShifToLeftChild(BTreeNode<int> x, int i, BTreeNode<int> y, BTreeNode<int> z)
        {
            y.KeyCount++;
            y.Keys[y.KeyCount - 1] = x.Keys[i];
            x.Keys[i] = z.Keys[0];
            for (int j = 0; j < z.KeyCount-1; j++)
            {
                z.Keys[j] = z.Keys[j+1];
            }

            if (!z.IsLeaf)
            {
                y.Childs[y.KeyCount] = z.Childs[0];
                for (int j = 0; j < z.KeyCount; j++)
                {
                    z.Childs[j] = z.Childs[j+1];
                }
            }

            z.KeyCount--;
        }

        #endregion
    }
}

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