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The process of inserting a node into a binary tree. Insertion on internal nodes is slightly more complex than on leaf nodes. Say that the internal node is node A and that node B is the child of A. (If the insertion is to insert a right child, then B is the right child of A, and similarly with a left child insertion.)
A binary heap is a heap data structure that takes the form of a binary tree. Binary heaps are a common way of implementing priority queues. [1]: 162–163 The binary heap was introduced by J. W. J. Williams in 1964 as a data structure for implementing heapsort. [2] A binary heap is defined as a binary tree with two additional constraints: [3]
Fig. 1: A binary search tree of size 9 and depth 3, with 8 at the root. In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree.
To insert a new element, search the tree to find the leaf node where the new element should be added. Insert the new element into that node with the following steps: If the node contains fewer than the maximum allowed number of elements, then there is room for the new element. Insert the new element in the node, keeping the node's elements ordered.
When inserting a node into an AVL tree, you initially follow the same process as inserting into a Binary Search Tree. If the tree is empty, then the node is inserted as the root of the tree. If the tree is not empty, then we go down the root, and recursively go down the tree searching for the location to insert the new node.
A min-max heap is a complete binary tree containing alternating min (or even) and max (or odd) levels. Even levels are for example 0, 2, 4, etc, and odd levels are respectively 1, 3, 5, etc. We assume in the next points that the root element is at the first level, i.e., 0. Example of Min-max heap
One picks a representative for each tree and insert these into the x-fast trie. Finding the successor of k takes O(log log M) time. Inserting k into a balanced binary search tree that contains O(log M) elements also takes O(log log M) time. Splitting a binary search tree that contains O(log M) elements can be done in O(log log M) time.
To search for a given key value, apply a standard binary search algorithm in a binary search tree, ignoring the priorities. To insert a new key x into the treap, generate a random priority y for x. Binary search for x in the tree, and create a new node at the leaf position where the binary search determines a node for x should exist.