meaning of the title

A tree with nnn nodes defines the process of igniting a sequence of nnn length by selecting the smallest numbered leaf node at a time to delete and adding the number of that node to the end of the sequence.Now there are qqq operations, each time you can turn a node's number to the maximum, or ask a point which position in the burned sequence it gets, or ask who is the first of the two points.
n,q≤200000n,q\le200000n,q≤200000

Analysis

Obviously the third can be done by calling the second two times.
Assuming that the current combustion sequence is known, setting the maximum numbered point as the root of the tree, when a point is numbered to the maximum, it is not difficult to find that only the new root to all the positions on the old root path in the combustion sequence are adjusted to the last, while the relative positions of the remaining points are unchanged.
If we change an assignment operation to a stain on the root path, the answer to a point is the number of points with a smaller color than it and the number of points with the same color that were deleted before it.
Maintain the tree with LCT, a single stain is equivalent to an access access access operation, and a weight tree array is equivalent to the size of the right subtree after splaysplaying the point to the root.
Time Complexity O(nlogn)O(nlogn)O(nlogn).

Code

#include<iostream>
#include<cstdio>
#include<cstdlib>
#include<cstring>
#include<algorithm>
#include<queue>

const int N=200005;
const int maxn=200000;

int n,q,cnt,last[N],deg[N],rk[N],c1[N],c2[N],tim;
struct edge{int to,next;}e[N*2];
struct tree{int l,r,fa,s,col;bool rev;}t[N];
std::priority_queue<int> que;

void addedge(int u,int v)
{
	e[++cnt].to=v;e[cnt].next=last[u];last[u]=cnt;
	e[++cnt].to=u;e[cnt].next=last[v];last[v]=cnt;
}

void pre()
{
	for (int i=1;i<=n;i++) if (deg[i]==1) que.push(-i),deg[i]=-1;
	for (int i=1;i<=n;i++)
	{
		int x=-que.top();que.pop();
		rk[x]=i;
		for (int j=last[x];j;j=e[j].next) if (deg[e[j].to]>0)
		{
			deg[e[j].to]--;
			if (deg[e[j].to]<=1) que.push(-e[j].to),deg[e[j].to]=-1;
		}
	}
}

void dfs(int x,int fa)
{
	t[x].s=1;
	for (int i=last[x];i;i=e[i].next) if (e[i].to!=fa) t[e[i].to].fa=x,dfs(e[i].to,x);
}

int find(int *c,int x)
{
	int ans=0;
	while (x) ans+=c[x],x-=x&(-x);
	return ans+c[0];
}

void ins(int *c,int x,int y)
{
	if (!x) c[0]+=y;
	else while (x<=maxn) c[x]+=y,x+=x&(-x);
}

bool is_root(int x)
{
	return x!=t[t[x].fa].l&&x!=t[t[x].fa].r;
}

void updata(int x)
{
	t[x].s=t[t[x].l].s+t[t[x].r].s+1;
}

void remove(int x)
{
	if (!is_root(x)) remove(t[x].fa);
	if (!t[x].rev) return;
	std::swap(t[x].l,t[x].r);t[x].rev=0; 
	if (t[x].l) t[t[x].l].rev^=1;
	if (t[x].r) t[t[x].r].rev^=1;
}

void rttl(int x)
{
	int y=t[x].r;t[y].col=t[x].col;
	t[x].r=t[y].l;t[t[y].l].fa=x;
	if (x==t[t[x].fa].l) t[t[x].fa].l=y;
	else if (x==t[t[x].fa].r) t[t[x].fa].r=y;
	t[y].fa=t[x].fa;
	t[y].l=x;t[x].fa=y;
	updata(x);updata(y);
}

void rttr(int x)
{
	int y=t[x].l;t[y].col=t[x].col;
	t[x].l=t[y].r;t[t[y].r].fa=x;
	if (x==t[t[x].fa].l) t[t[x].fa].l=y;
	else if (x==t[t[x].fa].r) t[t[x].fa].r=y;
	t[y].fa=t[x].fa;
	t[y].r=x;t[x].fa=y;
	updata(x);updata(y);
}

void splay(int x)
{
	remove(x);
	while (!is_root(x))
	{
		int p=t[x].fa,q=t[p].fa;
		if (is_root(p))
		{
			if (x==t[p].l) rttr(p);
			else rttl(p);
			break;
		}
		if (x==t[p].l)
			if (p==t[q].l) rttr(q),rttr(p);
			else rttr(p),rttl(q);
		else
			if (p==t[q].r) rttl(q),rttl(p);
			else rttl(p),rttr(q);
	}
}

void access(int x)
{
	int y=0;
	while (x)
	{
		splay(x);
		if (t[x].r) t[t[x].r].col=t[x].col;
		ins(c1,t[x].col,-t[x].s+t[t[x].r].s);
		if (!t[x].col) ins(c2,rk[x],1);
		t[x].r=y;updata(x);
		y=x;x=t[x].fa;
	}
	t[y].col=tim;
	ins(c1,tim,t[y].s);
}

void make_root(int x)
{
	access(x);splay(x);t[x].rev^=1;
}

int query(int x)
{
	splay(x);
	if (!t[x].col) return rk[x]-find(c2,rk[x]);
	else return find(c1,t[x].col-1)+t[t[x].r].s+1;
}

int main()
{
	scanf("%d%d",&n,&q);
	for (int i=1;i<n;i++)
	{
		int x,y;scanf("%d%d",&x,&y);
		addedge(x,y);
		deg[x]++;deg[y]++;
	}
	pre();
	dfs(n,0);
	c1[0]=n;
	while (q--)
	{
		char ch[10];scanf("%s",ch);
		if (ch[0]=='u')
		{
			int x;scanf("%d",&x);
			tim++;
			make_root(x);
		}
		else if (ch[0]=='w')
		{
			int x;scanf("%d",&x);
			printf("%d\n",query(x));
		}
		else
		{
			int x,y;scanf("%d%d",&x,&y);
			printf("%d\n",query(x)<query(y)?x:y);
		}
	}
	return 0;
}