#### Lowest Common Ancestor

D
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```# https://en.wikipedia.org/wiki/Lowest_common_ancestor

from __future__ import annotations

from queue import Queue

def swap(a: int, b: int) -> tuple[int, int]:
"""
Return a tuple (b, a) when given two integers a and b
>>> swap(2,3)
(3, 2)
>>> swap(3,4)
(4, 3)
>>> swap(67, 12)
(12, 67)
"""
a ^= b
b ^= a
a ^= b
return a, b

def create_sparse(max_node: int, parent: list[list[int]]) -> list[list[int]]:
"""
creating sparse table which saves each nodes 2^i-th 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 lowest_common_ancestor(
u: int, v: int, level: list[int], parent: list[list[int]]
) -> int:
# 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] not in [0, 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
level: list[int],
parent: list[list[int]],
max_node: int,
graph: dict[int, list[int]],
root: int = 1,
) -> tuple[list[int], list[list[int]]]:
"""
sets every nodes direct parent
parent of root node is set to 0
calculates depth of each node from root node
"""
level[root] = 0
q: Queue[int] = 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() -> None:
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: dict[int, list[int]] = {
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 = breadth_first_search(level, parent, max_node, graph, 1)
parent = create_sparse(max_node, parent)
print("LCA of node 1 and 3 is: ", lowest_common_ancestor(1, 3, level, parent))
print("LCA of node 5 and 6 is: ", lowest_common_ancestor(5, 6, level, parent))
print("LCA of node 7 and 11 is: ", lowest_common_ancestor(7, 11, level, parent))
print("LCA of node 6 and 7 is: ", lowest_common_ancestor(6, 7, level, parent))
print("LCA of node 4 and 12 is: ", lowest_common_ancestor(4, 12, level, parent))
print("LCA of node 8 and 8 is: ", lowest_common_ancestor(8, 8, level, parent))

if __name__ == "__main__":
main()
```