Trees

For large amounts of input, the linear access time of linked lists is prohibitive. In this chapter, we look at a simple data structure for which the running time of most operations is O(logN) on average.

In this chapter,  we will :

• see how trees are used to implement the file system of several popular operatiing systems.
• see how trees can be used to evaluate arithmetic expressions.
• Show how to use trees to support searching operations in O(logN) average time, and how to refine these ideas to obtain O(lognN) worst-case bounds we will also see how to implement these operations when the data are stored on a disk.

 

Preliminaries

     

     A tree can be defined in several ways. one natural way to define a tree is recursively. A tree is a collection of nodes. The collection can be empty; otherwise, a tree consists of a disinguished node r , called the root, and zero or more nonempty (sub)trees T1,T2,..,Tk, each of whose roots are connected by a directed edge from r.

 

     The root of each subtree is said to be a child of r, and r is the parent of each subtree root.

 

     From the recursive defintion, we find that a tree is a collection of N nodes, one of which is the root, and N-1 edges. That there are N-1 edges follows from the fact that each edge connects some node to its parent , and every node except the root has one parent.

 

 

     Leaves: the node in a tree has no child

     Sibling : the nodes have the same parent.

     Path : define as a sequence of nodes n1,n2,...,nk such that ni is the parent of ni+1 for 1<= i < k.

     The Length of Path : is the number of edges on the path, namely k-1.

 

     Depth : for any node ni, the depth of ni is the length of the unique path from  the root to ni. Thus, the root is at depth 0.

     Height : The height of ni is the length of the longest path from ni to a leaf.

 

 

 

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