Given a N-ary tree with a value associated with each node, the task is to choose a subset of these nodes such that sum of chosen nodes is maximum under a constraint that no two chosen node in subset should be directly connected that is, if we have taken a node in our sum then we can’t take its any children in consideration and vice versa.
The above diagram selects the nodes with a deep green color to get the maximum value 25.
In this post, we will be discussing an approach using Dynamic Programming on Trees.
While solving the problem, there arise two cases:
- For a particular node, the maximum sum can be calculated by including the node itself along with nodes from its subtree.
- Or, the maximum sum is calculated by excluding the current node and including only the nodes from its subtree.
Let us assume:
- dp1[node] to be the maximum possible sum by choosing nodes from the subtree of this node and also including the node.
- And, dp2[node] to be the maximum possible sum by choosing nodes from the subtree of the node and not including the node itself.
The first case, if we include the current node, then its value is added and then we can not include any of its immediate children, hence the summation of dp2 of all the children will be taken into the count to compute dp1[node]. That is,
dp1[node] = tree[node] + sum(dp2[children1], dp2[children2], …)
The second case, if we do not include the current node, then its value is not added, but the children node can be taken or it cannot be taken, hence the summation of the maximum of both for all the children will be taken into count to compute dp2[node]. That is,
dp2[node] = tree[node] + sum(max(dp1[children1], dp2[children1]), max(dp1[children2], dp2[children2])…)
In the end, the final answer will be the maximum of dp1[root] and dp2[root].
Below is the implementation of the above approach:
Maximum sum: 25
GeeksforGeeks has prepared a complete interview preparation course with premium videos, theory, practice problems, TA support and many more features. Please refer Placement 100 for details
- Maximum sum of nodes in Binary tree such that no two are adjacent
- Find maximum among all right nodes in Binary Tree
- Sum of nodes at maximum depth of a Binary Tree | Set 2
- Sum of nodes at maximum depth of a Binary Tree
- Maximum sum of non-leaf nodes among all levels of the given binary tree
- Maximum sum of leaf nodes among all levels of the given binary tree
- Sum of nodes at maximum depth of a Binary Tree | Iterative Approach
- Maximum length cycle that can be formed by joining two nodes of a binary tree
- Maximum weighted edge in path between two nodes in an N-ary tree using binary lifting
- Queries to find the maximum Xor value between X and the nodes of a given level of a perfect binary tree
- Minimum and maximum node that lies in the path connecting two nodes in a Binary Tree
- Vertex Cover Problem | Set 2 (Dynamic Programming Solution for Tree)
- Construct XOR tree by Given leaf nodes of Perfect Binary Tree
- Total sum except adjacent of a given node in a Binary Tree
- Maximum sum from a tree with adjacent levels not allowed
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to firstname.lastname@example.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.