```
# https://en.wikipedia.org/wiki/Lowest_common_ancestor
# https://en.wikipedia.org/wiki/Breadth-first_search
import queue
def swap(a, b):
a ^= b
b ^= a
a ^= b
return a, b
# creating sparse table which saves each nodes 2^i-th parent
def creatSparse(max_node, parent):
j = 1
while (1 << j) < max_node:
for i in range(1, max_node + 1):
parent[j][i] = parent[j - 1][parent[j - 1][i]]
j += 1
return parent
# returns lca of node u,v
def LCA(u, v, level, parent):
# u must be deeper in the tree than v
if level[u] < level[v]:
u, v = swap(u, v)
# making depth of u same as depth of v
for i in range(18, -1, -1):
if level[u] - (1 << i) >= level[v]:
u = parent[i][u]
# at the same depth if u==v that mean lca is found
if u == v:
return u
# moving both nodes upwards till lca in found
for i in range(18, -1, -1):
if parent[i][u] != 0 and parent[i][u] != parent[i][v]:
u, v = parent[i][u], parent[i][v]
# returning longest common ancestor of u,v
return parent[0][u]
# runs a breadth first search from root node of the tree
# sets every nodes direct parent
# parent of root node is set to 0
# calculates depth of each node from root node
def bfs(level, parent, max_node, graph, root=1):
level[root] = 0
q = queue.Queue(maxsize=max_node)
q.put(root)
while q.qsize() != 0:
u = q.get()
for v in graph[u]:
if level[v] == -1:
level[v] = level[u] + 1
q.put(v)
parent[0][v] = u
return level, parent
def main():
max_node = 13
# initializing with 0
parent = [[0 for _ in range(max_node + 10)] for _ in range(20)]
# initializing with -1 which means every node is unvisited
level = [-1 for _ in range(max_node + 10)]
graph = {
1: [2, 3, 4],
2: [5],
3: [6, 7],
4: [8],
5: [9, 10],
6: [11],
7: [],
8: [12, 13],
9: [],
10: [],
11: [],
12: [],
13: [],
}
level, parent = bfs(level, parent, max_node, graph, 1)
parent = creatSparse(max_node, parent)
print("LCA of node 1 and 3 is: ", LCA(1, 3, level, parent))
print("LCA of node 5 and 6 is: ", LCA(5, 6, level, parent))
print("LCA of node 7 and 11 is: ", LCA(7, 11, level, parent))
print("LCA of node 6 and 7 is: ", LCA(6, 7, level, parent))
print("LCA of node 4 and 12 is: ", LCA(4, 12, level, parent))
print("LCA of node 8 and 8 is: ", LCA(8, 8, level, parent))
if __name__ == "__main__":
main()
```