Dog Snacks

Gildong is playing with his dog, Badugi. They are at a park that has $n$ intersections and $n-1$ bidirectional roads. Each road is $1$ meter long and connects two intersections. The intersections are numbered from $1$ to $n$, and for every $a$ and $b$ ($1\le a,b\le n$), Badugi can get from intersection $a$ to intersection $b$ using some set of roads.

Gildong has put one snack at every intersection of the park. He gives Badugi a mission to eat all of the snacks. Badugi starts at intersection $1$ and moves by the following rules:

• Badugi looks for the snacks closest to him. Distance is the length of the shortest path from his current location to the intersection with the snack. His sense of smell reaches only $k$ meters, so he can find only snacks at distance $k$ or less. If he finds no such snack, he fails the mission.
• Among the snacks he can smell from his current location, he picks the one that minimizes the distance he must travel. If several snacks tie, he picks any one of them.
• He repeats this until he has eaten all $n$ snacks. Then he must find intersection $1$ again, and that intersection must be at distance $k$ or less from the last snack he ate. If he finds it, he completes the mission. Otherwise, he fails.

Gildong does not know $k$. Find the minimum value of $k$ that lets Badugi complete the mission when he moves optimally.

The first line contains the number of test cases $t$ ($1\le t\le 10^4$).

The first line of each test case contains one integer $n$ ($2\le n\le 2\cdot 10^5$), the number of intersections.

Each of the next $n-1$ lines contains two integers $u$ and $v$ ($1\le u,v\le n$, $u\ne v$), meaning there is a road between intersections $u$ and $v$. All roads are bidirectional and distinct.

It is guaranteed that for each test case, for every $a$ and $b$ ($1\le a,b\le n$), it is possible to get from intersection $a$ to intersection $b$ using roads. The sum of $n$ over all test cases does not exceed $2\cdot 10^5$.

For each test case, print one integer: the minimum value of $k$ such that Badugi can complete the mission.

In the first case, Badugi can complete the mission with $k=2$ by moving as follows:

1. Badugi starts at intersection $1$. The closest snack is at intersection $1$, so he eats it.
2. The next closest snacks are at intersection $2$ and intersection $3$. Suppose he picks intersection $2$. He moves $1$ meter to intersection $2$ and eats the snack.
3. The only remaining snack is at intersection $3$. He moves along $2$ roads to get there.
4. After eating it, he returns to intersection $1$, which is $1$ meter away. He completes the mission.

In the second case, the only possible sequence of moves is $1 \to 2 \to 3 \to 4 \to 1$. The distance from intersection $4$ to intersection $1$ is $3$, so $k$ must be at least $3$.

In the third case, Badugi can move as $1 \to 5 \to 6 \to 7 \to 8 \to 2 \to 3 \to 4 \to 1$. It can be shown that this is the only sequence that completes the mission with $k=3$.