COMP 2210 Module 14 - Search Trees

balance factor avl formula

BFm = Hr - Hl

a ______ tree will have near-optimal height for storing N nodes

balanced

deleting an element from an AVL tree

beginng atthe point of deletion, walk the reverse path back toward the root, recalculating balance factors stop at the first (lowest) node that has a balance factor of +-2. this node roots the 3-node neighborgood that will be rotated

avl search tree

binary search tree in which the heights of the left and right subtree of every node differ by at most one

a ______ node has exactly one _______

child parent

branches

connections between nodes, from parent to child. also called edges

preorder, inorder, and postorder are all _fs

d

case for deletion: the value is in a node with two non-empty subtrees

don't delete the node. find a replacement for this node's value and delete the node containing the replacement

ternary tree

each node can have at most 3 children

red black tree 4 rules

each node is either red or black the root and all empty trees are black all paths from the root to an empty tree contain the same number of black nodes a red node can't have a red child

binary search tree order

everything to the left is less everything to the right is greater than

a ______ node has no _______

leaf children

height

longest path from a given node to a descendant leaf

2-4 tree

m-way each non-leaf node must have at leats two non-empty subtrees

depth

measures the distance of a given node from the top of the tree. essentially level

descendant

node x is a descendant of node y if there is a path from y to x

ancestor

node x is an ancestor of rode y if there is a pth from x to y

a tree is composed of _____ and ______

nodes branches

binary search tree searching

o(height) height = n if tall and narrow height = logn if short and wide

worse case binary tree insert

o(height) inserting it as a child of the lowet leaf

height is

o(logn) for full, complete, balanced?

list is a _____ ________ structure because it defines linear relationships between elements: predecessor, successor

one dimensional

a _______ node has one or more ________

parent children

node

place in the tree where the elements are stored

multiple ________ operations may be required per deletion, so the reverse walk must go to the root each time

rebalancing

a ___ _____ is used like a filler. it allows a red-black tree to obey rules 1, 2, and 3 without being a perfect triangle (full)

red node

without ____ ______, red-black trees could only be full

red nodes

the _____ node has no _____

root parent

case for deletion: the value to delete is in a leaf node

set the parent's pointer to this node to null

case for deletion: the value to delete is in a node with exactly one non-empty subtree

set the parent's pointer to this node to this node's child

incremental balancing

the insert and delete algorithms are modified to ensure that the tree is balanced after each operation that could affect its height. trees that use incremental balancing are called self-balancing BSTs

path length

the number of nodes on the path

periodic balancing

the tree grows naturally without regard for its height or shape. at some interval (ether when a height threshold is reached or just on demand), the tree is reorganized to reduce its height to o(log n). the cost of this operation is amortized over many inserts and deletes

random adds

tree should be relatively flat

random removes

tree structure is bad. search and add becomes root of N

tree is a ____ __________ structure because it defines hierarchical relationships among elements: parent, child

two dimensional

m-way trees are different because they grow ___

up

bfs strategy

visit the nodes level by level

binary tree inorder

LNR left node right

binary tree post order

LRN left right node

binary tree preorder

NLR node left right

tree

a collection in which the elements are arranged in a hirearchy

path length note

a path is sometimes defined as a sequence of edges instead of nodes so path length is sometimes counted differently

path

a sequence of nodes from one node to another node, going from parent to child

balanced

a tree is balanced if for each node, its sub trees have similar heights

complete

a tree is complete if it is full to the next to last level, and the leaves on the lowest level are "left justified"

multi-way search tree

a tree of order m > 2 in which the search property (total order) holds on every node and in which all leaves are at the same depth

subtree

a tree within a larger tree, rooted at the given node x. the subtree consists of x and all descents of x. the tree itself is a subtree

full tree

all leaves have the same depth and every parent node has the maximum number of children