Trees : Part 1. Section 4.1. Theory and Terminology. A Tree? A Tree? Theory and Terminology. Theory and Terminology
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1 Trees : Part Section. () (2) Preorder, Postorder and Levelorder Traversals Definition: A tree is a connected graph with no cycles Consequences: Between any two vertices, there is exactly one unique path A Tree? A Tree? Definition: A ed tree is a graph G such that: G is connected G has no cycles G has exactly one vertex called the of the tree Consequences The depth of a vertex v is the length of the unique path from to v G can be arranged so that the is at the top, its neighboring vertices are vertices of depth, and so on The set of all vertices of depth k is called level k of the tree
2 A Rooted Tree Rooted Tree: Recursive definition depth = height = 3 height = 2 A graph with N nodes and N - edges Graph has one r Zero or more non-empty sub-trees, each of whose is connected to r by an edge. height = height = 0 Every node except the has one parent Definition: A descending path in a ed tree is a path, whose edges go from a vertex to a deeper vertex Consequences: A unique path from the to any vertex is a descending path The length of this path is the depth of the vertex depth = depth = Definition: If there is a descending path from v to v 2, v is an ancestor of v 2, and v 2 is a descendant of v. Suppose v is a vertex of depth k: Any vertex that is adjacent to v must have depth k - or k +. depth =
3 Suppose v is a vertex of depth k: Vertices adjacent to v of depth k + are called children of v. Suppose v is a vertex of depth k: If k > 0, there is exactly one vertex of depth k that is adjacent to v in the graph. This vertex is called the parent of v. depth = depth = Definitions A vertex with no children is called a leaf Definitions Depth of a vertex v is its distance from the. Height of a vertex v is the distance of the longest path from v to one of its descendant leaves. The height of a tree is the maximum depth of its vertices depth = height depth = Definitions The is the only vertex of depth 0. The has no parent. Example of ed tree Which are the parent nodes? Which are the child nodes? Which are the leaves? What is the height and depth of the tree? What is the height and depth of node E? Node F? depth =
4 Overview of Tree Implementation Each node points to Its first child Its next sibling Back to its parent (optional) What could be an alternate representation? Tree Traversals Definition: A traversal is the process for visiting all of the vertices in a tree Often defined recursively Each kind corresponds to an iterator type Iterators are implemented non-recursively Preorder Traversal Visit vertex, then visit child vertices (recursive definition) Depth-first search Begin at Visit vertex on arrival Implementation may be recursive, stackbased, or nested loop Preorder Traversal Preorder Traversal of UNIX Directory Tree Postorder Traversal Visit child vertices, then visit vertex (recursive definition) Depth-first search Begin at Visit vertex on departure Implementation may be recursive, stackbased, or nested loop
5 Postorder Traversal Postorder Traversal Calculating Size of Directory 7 Levelorder Traversal Visit all vertices in level, starting with level 0 and increasing Breadth-first search Begin at Visit vertex on departure Only practical implementation is queuebased Levelorder Traversal 7 Tree Traversals Preorder: depth-first search (possibly stack-based), visit on arrival Postorder: depth-first search (possibly stack-based), visit on departure Levelorder: breadth-first search (queuebased), visit on departure 5
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