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Lecture 4
Lecture 4

Persistent Binary Search Trees
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Balancing Trees
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Data Structures and Algorithms
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Trees and Binary Search Trees Dynamic data structures Tree: Tree:
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... - All the values in the left subtree are smaller than the value at X. - All the values in the right subtree are larger than the value at X. ...
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... the left or right of the current node level 2 ...
CS2351 Data Structures
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... binary tree. Now look at the nodes that make up the linked list. The nodes have the same type structure -- they each contain an element and two pointers. The only difference is that in the tree, the two pointers are labeled "small" and "large" while in the list they are labeled "previous" and "next" ...
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Scapegoat tree
Scapegoat tree

... • All n keys are stored in the leaves • Internal nodes store 1, 2, or 3 values to direct searches to the correct subtree • Searches take O(h) = O(log n) time • Theorem: A 2-4 tree supports the operations find(x), add(x), and remove(x) in O(log n) time per operation ...
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...  There are 3 ways to traverse a tree, that is, to visit every node:  Preorder traversal: visit the current node, then traverse its left subtree, then its right subtree  Postorder traversal: traverse the left subtree, then the right subtree, then visit the current node  Inorder traversal: travers ...
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... 3) Every node except root must contain at least t-1 keys. Root may contain minimum 1 key. 4) All nodes (including root) may contain at most 2t – 1 keys. 5) Number of children of a node is equal to the number of keys in it plus 1. 6) All keys of a node are sorted in increasing order. The child betwee ...
Red-black trees
Red-black trees

< 1 ... 54 55 56 57 58 59 60 61 >

Red–black tree

A red–black tree is a binary search tree with an extra bit of data per node, its color, which can be either red or black. The extra bit of storage ensures an approximately balanced tree by constraining how nodes are colored from any path from the root to the leaf. Thus, it is a data structure which is a type of self-balancing binary search tree.Balance is preserved by painting each node of the tree with one of two colors (typically called 'red' and 'black') in a way that satisfies certain properties, which collectively constrain how unbalanced the tree can become in the worst case. When the tree is modified, the new tree is subsequently rearranged and repainted to restore the coloring properties. The properties are designed in such a way that this rearranging and recoloring can be performed efficiently.The balancing of the tree is not perfect but it is good enough to allow it to guarantee searching in O(log n) time, where n is the total number of elements in the tree. The insertion and deletion operations, along with the tree rearrangement and recoloring, are also performed in O(log n) time.Tracking the color of each node requires only 1 bit of information per node because there are only two colors. The tree does not contain any other data specific to its being a red–black tree so its memory footprint is almost identical to a classic (uncolored) binary search tree. In many cases the additional bit of information can be stored at no additional memory cost.
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