[BZOJ1005] Prufer Series + permutation combination

The prufer sequence of a tree

Each time you find the smallest leaf node in the remaining tree (which is a node with a degree of 1 for a tree without root), delete it.

The parent node (that is, the unique point connected to it) is added to the prufer sequence.

A tree corresponding to a prufer sequence

G sets start as empty set

Set the current processing to the Prufer sequence of item I, found in the G collection does not appear and in Prufer[i. N-2] The smallest node of the label that is not present, set it to U.

Add u to the set G and the U to Prufer[i] with an edge.

Finally, an edge is attached between two points that are still not present in the G collection (one must be N).

To think about why you can restore it to the original tree.

First of all, we can consider the operation of adding the G set as deleting the leaf node.

No representation of this number in the G set is still in the tree and has not been deleted.

Again prufer[i. N-2] does not appear to represent the current degree of the node is 1, that is, the leaf node.

So the finding of U is actually equivalent to the smallest leaf node of the label.

Obviously, the tree and Prufer series one by one correspond.

That is, there is only one prufer sequence for a tree.

For a prufer sequence, only one tree can be restored.

Some properties

Each prufer sequence has n-2 entries (n is the number of nodes)

For a node with a degree of x, it appears x-1 times in the Prufer sequence

The prufer sequence that satisfies both of these conditions (of course each number cannot exceed N) must be able to form a valid tree

　With this idea, you can begin to do the problem.

That is, the number of possible Prufer series.

into a permutation and combination problem

For the CNT has a degree limit of the node, its a[i]-1 and we set to sum, then the prufer sequence of n-2 in the space to select the sum position of the CNT dot

For the rest of the n-sum-2, you can place arbitrary points without a degree limit.

　　(n-2)!* (n-cnt) n-sum-2/(∏ (a[i]-1)!* (n-sum-2)!)

　　The decomposition of the mass factor can be used to solve the above-mentioned formula, and a faster method can be used for the decomposition of factorial.

Final high-precision multiplication.

Program bzoj1005;Constmaxn=1010; tt=10000; type arr=array[-1.. Maxn]of Longint;varN,cnt,sum,i,j:longint; a,p,w:array[-1..    Maxn]of Longint; vis:array[-1..    Maxn]of Boolean;    Ans:arr; SS:string;p rocedure printf;begin writeln (0); Halt;end;procedure build;varI,j:longint;begin Fillchar (Vis,sizeof(VIS),true); p[0]:=0;  fori:=2To MAXN DobeginifVis[i] THEN BEGIN Inc. (p[0]); p[p[0]]:=i;        End  forj:=1To p[0] DobeginifI*P[J]&GT;MAXN Then Break; Vis[i*p[j]]:=false; ifI mod p[j]=0Then Break;    End End;end;procedure work (x,y:longint);vari,xx,tot:longint;begin XX:=x;  fori:=1To p[0] Dobegin x:=xx;tot:=0;  whileX<>0  DoBegin Inc (Tot,x Div P[i]); X:=x Div p[i];        End Inc (W[i],tot*y); End;end;procedure Work2 (x,y:longint);varI,tot:longint;begin fori:=1To p[0] Do ifX mod p[i]=0Then begin tot:=0;  whileX mod p[i]=0  DoBegin Inc (TOT); x:=x Div p[i];        End Inc (W[i],tot*y); End;end;function Mul (a:arr;b:longint): arr;varC:arr; I:longint;begin Fillchar (c,sizeof(c),0); c[0]:=a[0];  fori:=1To c[0] DoBegin Inc (C[I],A[I)*b); Inc (C[i+1],c[i] div tt); C[i]:=C[i] MoD tt;    End ifc[c[0]+1]<>0Then Inc (c[0]); Exit (c); End;begin//assign (input, ' bzoj1005.in '); reset (input); //assign (output, ' bzoj1005.out '); rewrite (output);READLN (n); CNT:=0; sum:=0;  fori:=1to n Dobegin READLN (A[i]); ifA[i]>0Then Inc (CNT)Else            ifa[i]=0then printf; ifA[i]>0Then Inc (sum,a[i]-1);    End Fillchar (W,sizeof(W),0); ifsum>n-2then printf;    Build Work (n-2,1); Work (n-2-sum,-1); WORK2 (n-cnt,n-sum-2);  fori:=1to n Do ifA[i]>0Then work (a[i]-1,-1); ans[0]:=1; ans[1]:=1;  fori:=1To p[0] Do         forj:=1To W[i] Doans:=Mul (Ans,p[i]); Write (ans[ans[0]]);  fori:=ans[0]-1Downto1  Dobegin STR (ANS[I],SS);  forJ:=length (ss) +1To4  DoWrite0);    Write (Ans[i]); End;end.

[BZOJ1005] Prufer Series + permutation combination