Given an N-ary tree T of N nodes, the task is to calculate the longest path between any two nodes(also known as the diameter of the tree).
Different approaches to solve these problems have already been discussed:
In this post, we will be discussing an approach which uses Dynamic Programming on Trees.
There are two possibilities for the diameter to exist:
- Case 1: Suppose the diameter starts from a node and ends at some node in its subtree. Let’s say that there exist a node x such that the longest path starts from node x and goes into its subtree and ends at some node in the subtree itself. Let’s define this path length by dp1[x].
- Case 2: Suppose the diameter or the longest path starts in subtree of a node x, passes through it and ends in it’s subtree. Let’s define this path by dp2[x].
If for all nodes x, we take a maximum of dp1[x], dp2[x], then we will get the diameter of the tree.
For the case-1, to find dp1[node], we need to find the maximum of all dp1[x], where x is the children of node. And dp1[node] will be equal to 1 + max(dp1[children1], dp1[children2], ..).
For the case-2, to find dp2[node], we need to find the two maximum of all dp1[x], where x is the children of node. And dp2[node] will be equal to 1 + max 2 of(dp1[children1], dp1[children2], ..).
We can easily run a DFS and find the maximum of both dp1[node] and dp2[node] for every to get the diameter of the tree.
Below is the implementation of the above approach:
# Python3 program to find diameter
# of a tree using DFS.
# Function to find the diameter of the
# tree using Dynamic Programming
def dfs(node, parent, dp1, dp2, adj):
# Store the first maximum and secondmax
firstmax, secondmax = -1, -1
# Traverse for all children of node
for i in adj[node]:
if i == parent:
# Call DFS function again
dfs(i, node, dp1, dp2, adj)
# Find first max
if firstmax == -1:
firstmax = dp1[i]
elif dp1[i] >= firstmax: # Secondmaximum
secondmax = firstmax
firstmax = dp1[i]
elif dp1[i] > secondmax: # Find secondmaximum
secondmax = dp1[i]
# Base case for every node
dp1[node] = 1
if firstmax != -1: # Add
dp1[node] += firstmax
# Find dp
if secondmax != -1:
dp2[node] = 1 + firstmax + secondmax
# Return maximum of both
return max(dp1[node], dp2[node])
# Driver Code
if __name__ == “__main__”:
n, diameter = 5, -1
adj = [ for i in range(n + 1)]
# create undirected edges
dp1 =  * (n + 1)
dp2 =  * (n + 1)
# Find diameter by calling function
print(“Diameter of the given tree is”,
dfs(1, 1, dp1, dp2, adj))
# This code is contributed by Rituraj Jain
Diameter of the given tree is 4
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Improved By : rituraj_jain