# Day2 Balance Tree notes, day2 Balance Tree

Source: Internet
Author: User

Day2 Balance Tree notes, day2 Balance Tree
Operations not supported by the line segment tree: delete, insert

Common Balance Tree
```Treap slow | well-written sbt (size-balanced tree) Fast and easy-to-write | incomplete function rbt red/black tree extremely fast |Very hard to writeThe preceding Operations Support inserting and deleting O (NlogN)
Splay is particularly slow .. ≈ O (sqrt (N) is not easy to write and has powerful functions.```

Persistent Treap
`The Balance Tree must be a binary tree.The left son must have fewer elements than him.The right son must be larger than the current nodeThe middle-order traversal must be sorted`

Each recursive query

Small -- "left

Big -- "right

Disadvantages: the depth may be very deep --> the cost is very high

Treap = Tree + heap

Treap: stores two values: [key, val]

Val: The value inserted each time to satisfy the nature of the Balance Tree.

Key: meets the heap nature. rand must be at the logN level in depth.

Merge (p1, p2): merges the Treap with p1 as the root and the Treap with P2 as the root into a Treap. The maximum value of p1 should be <= the minimum value of P2

Split (p, k): split the Treap with p as the root into two treaps, one with k numbers and the other with n-k numbers. k is the first k smaller.

Insert: divide the tree into two parts: x and y. Then, regard the new node a as a tree. Merge the tree with x first. After the merge, merge the entire tree with y.

Delete:

Merge implementation

First find the largest key, compare p1, p2

• If p1 is large
`P1 is the root, p2 must be on the Right of p1,P1.L = p1.LP1.r = merge (p2, p1.r)`
• If p2
`p2.r=p2.rp2.L=merge(p2.L,p1)`

Merge returns the root node.

Split implementation

Size: number of nodes in the subtree

When k <= p. L. size-> split (p. L, k)->If p1 is set to a useful subtree, merge (p2, p. r) is used directly, and p2 is used as the left child of p.

Returns p. L + p, p. r when k = p. L. size + 1.

When k> p. l. size + 1-> split (p. r, k-p.L.size-1)-> set p2 as a useful subtree, direct merge (p, p1), take p1 as the right child of p

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