Levelwise Alternating OR and XOR operations in Segment Tree

A Levelwise OR/XOR alternating segment tree is a segment tree, such that at every level the operations OR and XOR alternate. In other words at Level 1 the left and right sub-trees combine together by the OR operation i.e Parent node = Left Child OR Right Child and on Level 2 the left 

and right sub-trees combine together by the XOR operation i.e Parent node = Left Child XOR Right Child
Such a type of Segment tree has the following type of structure: 
 

The operations (OR) and (XOR) indicate which operation was carried out to merge the child node
Given 2^N leaf nodes, the task is to build such a segment tree and print the root node 

Examples:  

Input : Leaves = {1, 6, 3, 7, 5, 9, 10, 4}
Output : Value at Root Node = 3
Explanation : The image given above shows the 
segment tree corresponding to the given
set leaf nodes.

This is an Extension to the Classical Segment Tree where we represent each node as an integer and the parent node is built by first building the left and the right subtrees and then combining the results of the left and the right children into the parent node. This merging of the left and right children into the parent node is done by consistent operations, For example, MIN()/MAX() in Range Minimum Queries, Sum, XOR, OR, AND, etc. By consistent operations, we mean that this operation is performed to merge any node’s left and right child into the parent node by carrying the operation say OP on their results, where OP is a consistent operation.

In this Segment tree, we carry two operations that are -: OR and XOR.
Now we build the Segment tree in a similar fashion as we do in the classical version, but now when we recursively pass the information for the sub-trees we will also pass information regarding the operation to be carried out at that level since these operations alternate levelwise. It’s important to note that a parent node when calls its left and right children the same operation information is passed to both the children as they are on the same level.

let’s represent the two operations i.e OR and XOR by 0 and 1 respectively. Then if at Level i we have an OR operation at Level (i + 1) we will have an XOR operation. There if Level i has 0 as operation then level (i + 1) will have 1 as operation

Operation at Level (i + 1) = ! (Operation at Level i)
where,
Operation at Level i ? {0, 1}

The parent-child relationship for a segment tree node is shown in the below image: 

Now, we need to look at the operation to be carried out at root node. 
 

If we carefully look at the image then it would be easy to figure out that if the height of the tree is even then the root node is a result of XOR operation of its left and right children else a result of OR operation. 

Value at Root Node = 3

Time Complexity for tree construction is O(n). There are total 2n-1 nodes, and value of every node is calculated only once in tree construction.

We can also perform Point Updates in a similar manner. If we get an update to update the leaf at index i of the array leaves then we traverse down the tree to the leaf node and perform the update. While coming back to the root node we build the tree again similar to the build() function by passing the operation to be performed at every level and storing the result of applying that operation on values of its left and right children and storing the result into that node.
Consider the following Segment tree after performing the update, 
Leaves[0] = 13 

Now the updated segment tree looks like this:
 

Here the nodes in black denote the fact that they were updated

Old Value at Root Node = 3
New Value at Root Node = 11

The time complexity of update is also O(Logn). To update a leaf value, we process one node at every level and number of levels is O(Logn).