Trees CS 105

L9: Trees Slide 2 Definition The Tree Data Structure stores objects (nodes) hierarchically nodes have parent-child relationships operations include accessing the parent or children of a given node A tree T is a set of nodes such that there is a distinguished node r (called the root) of T that has no parent each node v of T except r has a parent node

L9: Trees Slide 3 Visualizing a Tree N AP T M I O R G Root Child

L9: Trees Slide 4 Sample Uses A company’s organizational structure Family tree The Java class hierarchy O/S directory structure Book (parts, chapters, sections) Arithmetic expressions Web page links Priority queues and search trees

L9: Trees Slide 5 Tree Terminology Root the only node that has no parent Leaf (External node) node that has no children Internal node node that has at least one child Siblings nodes that have a common parent

L9: Trees Slide 6 More Tree Terminology Ancestor recursive definition: ancestors of a node v are v itself and the ancestors of its parent proper ancestors: ancestors excluding itself Descendant v is a descendant of u if u is an ancestor of v Subtree of T rooted at v set of all descendants of v

L9: Trees Slide 7 Even More Terminology Ordered Tree children of a node have a strict linear order Binary Tree an ordered tree where nodes have at most two children (left and right) Depth and Height of a node depth: distance from node to root height: from node to its farthest descendant

L9: Trees Slide 8 Tree Implementations Array-based implementation elements (or references to them) are stored in an array parent-child relationships derived from indices Linked implementation elements stored in nodes nodes contain pointers to parent and children

L9: Trees Slide 9 Tree Operations Get the root node of the tree Go to parent or children from a given node Add a root to an empty tree Add a child to a node Remove a node (can impose that the node be a leaf, for simplicity) Get the element associated to a node Replace the element associated to a node Others …

L9: Trees Slide 10 Binary Tree operations For a binary tree, a node has at most two children and are ordered Could distinguish between the left and right child of a node Implementations are simpler for binary trees but many concepts apply to general trees

L9: Trees Slide 11 Implementing trees Array implementation Array indices are used to navigate from parent to children or from children to parent Node implementation Node contains data and references to children and parent

L9: Trees Slide 12 Data structures that use trees Priority queues Heap: tree structure (usually array-based) where the hierarchy imply the priorities of the elements Dictionary Binary Search Tree: tree structure (usually node-based) that facilitates “binary search” of the nodes