BST
What are the characteristics of a general tree
-either empty or consists of a finite set of nodes T -node R is called the root -The set T-{r} is partitioned into disjoint subsets, each of which is a general tree
The number of nodes, N, contained in a full binary tree of height H is...
2^h +1 -1
Add the following numbers, in the order given to a binary search tree. 3,14,15,20,25,33, 62,200 Is this tree balanced? Why not? What is the execution time for searching for a value in this? What is the height of the tree?
3 14 15 20 goes down into a linear tree Not balanced. Execution time is O(n) 7
Child
A node immediately below and directly connected to a given node
Interior Node
A node that has at least one child
Descendant
A node's children, its children's children, and so on, down to the leaves
Ancestor
A node's parent, its paren'ts parent, and so on, up to the root
In an expression tree, each leaf node represents what?
A numeric operand
why does the shape of a binary tree matter?
A vine-like tree would be a more expensive path if you are looking for a node and had to go through each node
What are the 3 common applications of binary trees
Heaps, binary search trees, expression trees
Is implementing a preorder better externally or internally?
Interally because you very rarely want to just traverse the tree, in most cases you want to accomplish something else.
Siblings
The children of a common parent
Height
The length of the longest path in the tree -the maximum level number among leaves in the tree
Edge/Branch/Link
The line that connects a parent to its child
Postorder Traversal
Traverses left subtree, traverse right subtree, and visit root node -LRP
Inorder traversal
Traverses left subtree, visits root node, and traverses right subtree -LPR
Children are viewed from left to right. True or False?
True
Preoder traversal
Visits root node, and then traverses left subtree and right subtree In similar way -PLR
Are traversals written recursively?
Yes
Can you move subtrees (entire sections of a tree) to a different position in the tree without affecting the lower levels of the hierarchy?
Yes
What is the height of a tree containing one node
Zero
A tree has the following properties..
-One node of the tree is designated as the root node. -Every node n, except the root node, is connected by an edge from exactly one other node p, where p is the parent of n -A unique path traverses from the root to each node -If each node in the tree has a maximum of two children, it is a binary tree
Why use a tree?
-Parse tree -used for decision making -can support logarithmic searches and insertions -
Different shapes of binary trees
-Unbalanced -Perfectly balanced -Complete -Full
Heap property
constraint on the order of nodes
Operands of higher precedence usually appear near....
the bottom of the tree, unless override in source expression by parentheses
What is the root depth or level
the root is 0
In a binary tree, each node has at most...
two children, left and right child
What is the height of an empty tree
-1
Level order traversal
Beginning with level 0, visits the nodes at each level in left-to-right order
Heap sort
builds a head from data and repeatedly removes the root item and adds it to the end of a list
In worst case shape of a binary tree, they become
linear and support linear searches
Max heap-
places larger nodes nearer to the root
Add the following numbers in the order given to a binary search tree. 45,67,22,100,75,12,11,64,30 What is the height of the tree from number 1? What is the height of the subtree rooted at the node holding the value 22? If you remove 67, what would replace it?
45 22 67 13 30 | 64 100 11 75 height=3, starts from 0 Height of subtree holding the value 22= 2 64 would replace 67 because it is the largest subtree node of 67
Create a binary search tree and insert the following elements in the following order: 8,10,3,5,7,9,1,2,0 What is the depth? How many leaves? Give the preorder traversal of this tree? Give the inorder traversal of this tree Give the post order traversal of this tree.
8 3 \ 10 1/ \ 5 9/ 0/ \2 \ 7 depth=3 leaves=4 preorder=8,3,1,0,2,5,7,10,9 inorder=0,1,2,3,5,7,8,9,10 postorder=0,2,1,7,5,3,9,10,8
Parent
A node immediately above and directly connected to a given node. A node can have only one parent
Can an expression tree be empty?
An expression tree is NEVER empty
Node
An item stored in a tree
Expression Tree is..
Another way to process expressions to build a parse tree during parsing
What is a parse tree
Describes the syntactic structure of a particular sentence in terms of its component parts
What are the 2 main characteristics of trees?
Each item can have multiple children -except for root, root has exactly one parent
Definition of a binary tree
Either empty or consists of a root plus a left subtree and a right subtree, each of which are binary tress
A binary tree can have more than two children coming off of it
False
is a file system structure a binary tree or general tree
General tree because it has more than 2 children per node
When to use inorder traversal
Good for visiting items in a BST sorted order
What is a node that has no children?
Leaf
What is the leftmost and rightmost child called?
Leftmost=first rightmost= last child
A bst imposes a sorted ordering on its nodes. Nodes in left subtree of a node are _ node Nodes in right subtree of a node are _ node
Nodes in left subtree of a node are < node Nodes in right subtree of a node are > node
When shape approaches that of a perfectly balanced binary tree, searches and insertions are ____ in the worst case
O( log n)
The max amount of work that it takes to access a given node in a full binary tree is..
O(log N)
Where are trees used in computer science?
Operating systems, graphics, database systems, and computer networkin
What are the 4 standard types of traversals for binary trees
Preorder, inorder, postorder, level order
With every traversal, where do we start?
Root node
Path length
The number of edges in a path
Path
The sequence of edges that connect a node and one of its descendants
Root
The topmost node in a tree. It is the only node without a parent
Subtree
The tree formed by considering a node and all its descendants
Min heap-
each node is <= to both of its children
Depth or level
equals the length of the path connecting it to the root.
The height, H, of a full binary tree with N nodes is..
log v2 (N +1) -1
A BST imposes a special ordering on the nodes in a binary tree, so as to support___
logarithmic searches and insertions
A node can have how many children..
multiple
In a tree, the ideas of predecessor and successor are replaced with those of ___ and ___
parent and child
A node can have additional information called...
payload
Heaps are also used to implement...
priority queues
In an expression tree, an interior node represents a ______ _____, consisting of an _____
represents a compound expression, consisting of an operator and its operands