Suppose the data for a node appears in component [ i ] of the array. In the actual implementation, you will need: The array itself A second instance variable keeps track of how much of the array is used The actual links between the nodes are not stored, but are determined by the formula. Can we also implement a non-complete binary tree using an array? Node representation of binary trees : Each node of a binary tree can be stored as an object of a binary tree node class.
The class contains private instance variables that are references to other nodes in the tree. An entire tree is represented as a reference to the root node. Binary Tree Implementation with Tree Nodes. The return value is a reference to the root of the new tree. Return value could be null. Traversal : the process of visiting all the nodes in a tree in a certain order. Pre-order traversal In-order traversal Post-order traversal Pre-order traversal : Process the root.
Process the nodes in the left subtree with a recursive call. Process the nodes in the right subtree with a recursive call. In-order traversal : Process the nodes in the left subtree with a recursive call. Process the root. Post-order traversal : Process the nodes in the left subtree with a recursive call. In a tree, leaf node is also called as ' Terminal ' node. In simple words, an internal node is a node with atleast one child.
In a tree data structure, nodes other than leaf nodes are called as Internal Nodes. The root node is also said to be Internal Node if the tree has more than one node.
Internal nodes are also called as ' Non-Terminal ' nodes. In simple words, the Degree of a node is total number of children it has. The highest degree of a node among all the nodes in a tree is called as ' Degree of Tree '. In a tree data structure, the root node is said to be at Level 0 and the children of root node are at Level 1 and the children of the nodes which are at Level 1 will be at Level 2 and so on In simple words, in a tree each step from top to bottom is called as a Level and the Level count starts with '0' and incremented by one at each level Step.
In a tree data structure, the total number of edges from leaf node to a particular node in the longest path is called as HEIGHT of that Node. In a tree, height of the root node is said to be height of the tree. In a tree, height of all leaf nodes is '0'. In a tree data structure, the total number of egdes from root node to a particular node is called as DEPTH of that Node.
There is only one root per tree and one path from the root node to any node. If the root node is at level 0, then its next child node is at level 1, its grandchild is at level 2, and so on.
Binary Search tree exhibits a special behavior. A node's left child must have a value less than its parent's value and the node's right child must have a value greater than its parent value. The code to write a tree node would be similar to what is given below. It has a data part and references to its left and right child nodes. Like real trees, we have the root , branches , and finally the leaves. The height of a tree is the length of the longest path to a leaf.
The depth of a node is the length of the path to its root. Now we will discuss a specific type of tree. We call it the binary tree. The first thing we need to keep in mind when we implement a binary tree is that it is a collection of nodes.
How do we implement a simple binary tree that initializes with these three properties? When we instantiate an object, we pass the value the data of the node as a parameter. Both are set to None.
We just have the node data. We will implement a method to insert a new node to the right and to the left. DFS explores a path all the way to a leaf before backtracking and exploring another path. Then go to the left child 3. Print it. Backtrack and go the right child 4. Backtrack to the root node and go to the right child 5.
Go to the left child 6. Backtrack and go to the right child 7. Now that we are familiar with this traversal algorithm, we will discuss types of DFS : pre-order , in-order , and post-order.
The result of the post order algorithm for this tree example is 3—4—2—6—7—5—1. BFS algorithm traverses the tree level by level and depth by depth. To implement a BFS algorithm, we use the queue data structure to help. An important property of a Binary Search Tree is that the value of a Binary Search Tree node is larger than the value of the offspring of its left child , but smaller than the value of the offspring of its right child.
What will we see here? We will insert new nodes, search for a value, delete nodes, and the balance of the tree. Imagine that we have an empty tree and we want to add new nodes with the following values in this order: 50, 76, 21, 4, 32, , 64, The powerful part of this algorithm is the recursion part, which is on line 9 and line Lines 11 and 15 are the ones that do the insertion for each child.
The algorithm that we will build now is about doing searches. For a given value integer number , we will say if our Binary Search Tree does or does not have that value. An important item to note is how we defined the tree insertion algorithm. First we have our root node.
All the left subtree nodes will have smaller values than the root node. And all the right subtree nodes will have values greater than the root node. Yeah, it works for these given values! Deletion is a more complex algorithm because we need to handle different cases.
For a given value, we need to remove the node with this value. Imagine the following scenarios for this node : it has no children , has a single child , or has two children. If the node we want to delete has no children, we simply delete it.
In this case, our algorithm needs to make the parent of the node point to the child node. If the node is the left child , we make the parent of the left child point to the child.
If the node is the right child of its parent, we make the parent of the right child point to the child. We will put this node with minimum value in the place of the node we want to remove.
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