Editorial

Perform a depth-first search starting from any vertex, for example, from vertex $1$. Find the vertex farthest from $1$ and denote it as vertex $a$. Then, start a depth-first search from vertex $a$ to find the vertex that is farthest from $a$, which we'll denote as vertex $b$. The distance between vertices $a$ and $b$ will then be the maximum possible and equal to the diameter of the tree.

Example

Consider the tree from the example.

The diameter of the tree is $3$. The maximum distance is achieved, for example, between vertices $2$ and $5$.

Exercise

Find the diameter of the tree.

Algorithm implementation

Store the input tree in the adjacency list $g$. Store the shortest distances from the source to each vertex in the array $dist$.

vector<vector<int>> g;
vector<int> dist;

The dfs function performs a depth-first search from vertex $v$. The shortest distance from the source to vertex $v$ is $d$. The parent of vertex $v$ in the depth-first search is $p$.

void dfs(int v, int d, int p = -1)
{

Store the shortest distance from the source to vertex $v$ in $dist[v]$.

  dist[v] = d;

Iterate over all edges $(v,to)$. For each son to of vertex $v$ (where $to$ is not a parent of $v$) initiate a depth-first search. The distance from the source to vertex $to$ will be $d+1$.

  for (int to : g[v])
    if (to != p) dfs(to, d + 1, v);
}

The main part of the program. Read the input data.

scanf("%d", &n);
g.resize(n + 1);
dist.resize(n + 1);

Construct an undirected tree.

for (i = 1; i < n; i++)
{
  scanf("%d %d", &u, &v);
  g[u].push_back(v);
  g[v].push_back(u);
}

Find the shortest distances from vertex $1$ to all other vertices. The vertex farthest from it is $a$.

dfs(1, 0, -1);
a = max_element(dist.begin(), dist.begin() + n + 1) – 
                dist.begin();

Find the shortest distances from vertex $b$ to all other vertices. The vertex farthest from it is $b$.

dfs(a, 0,  -1);
b = max_element(dist.begin(), dist.begin() + n + 1) – 
                dist.begin();

Print the diameter of the tree — the shortest distance from $a$ to $b$.

printf("%d\n", dist[b]);