1. size优化解树形背包

注意：该方法无法解决带权树形背包。

P3177 HAOI2015 树上染色

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#include<bits/stdc++.h>
using namespace std;
#define int long long
const int MAXn = 2e3;
const int MAXm = MAXn;
const int MAXk = 2e3;

template <typename T> 
inline void read(T &a) {
    register char c;while (c = getchar(), c < '0' || c > '9');register T x(c - '0');while (c = getchar(), c >= '0' && c <= '9') {x = (x << 1) + (x << 3) + (c ^ 48);}a = x;  
}
template <typename T, typename ...Argv>
inline void read(T &n, Argv &...argv) {
    read(n), read(argv...);
}

int head[MAXm + 10], cntnex, nex[MAXm * 2 + 10], to[MAXm * 2 + 10], wei[MAXm * 2 + 10];
inline void Insert(int u, int v, int w) {
    nex[++cntnex] = head[u];
    head[u] = cntnex;
    to[cntnex] = v;
    wei[cntnex] = w;
}

int n, m, fa[MAXn + 10], siz[MAXn + 10], d[MAXn + 10][MAXk + 10];
void Dfs(int cur) {
    siz[cur] = 1;
    for (int i = head[cur]; i; i = nex[i]) {
        if (to[i] == fa[cur]) continue;
        fa[to[i]] = cur;
        Dfs(to[i]);
        siz[cur] += siz[to[i]];
    }
}
void DfsDp(int cur) {
    d[cur][0] = d[cur][1] = 0;
    for (int i = head[cur]; i; i = nex[i]) {
        if (to[i] == fa[cur]) continue;
        DfsDp(to[i]);
        for (int j = min(m, siz[cur]); ~j; --j) {
            for (int k = 0; k <= min(j, siz[to[i]]); ++k) {
                d[cur][j] = max(d[cur][j], d[cur][j - k] + d[to[i]][k] + wei[i] * (k * (m - k) + (siz[to[i]] - k) * (n - siz[to[i]] - (m - k))));
            }
        }
    }
}

signed main() {
    memset(d, 0xc0, sizeof(d));
    read(n, m);
    if (n < m * 2) {
        m = n - m;
    }
    for (int i = 1, u, v, w; i < n; ++i) {
        read(u, v, w);
        Insert(u, v, w); Insert(v, u, w);
    }
    Dfs(1);
    DfsDp(1);
    printf("%lld\n", d[1][m]);
}

2. dfs序优化解树形背包

U189430 带权树形背包

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#include<bits/stdc++.h>
using namespace std;
const int MAXn = 5e3;
const int MAXW = 1e4;
const int MAXm = MAXn;

template <typename T>
inline void read(T &a) {
    char c;for (c = getchar(); (c < '0' || c > '9') && c != '-'; c = getchar());bool f = c == '-';T x = f ? 0 : (c ^ '0');for (c = getchar(); c >= '0' && c <= '9'; c = getchar()) {x = x * 10 + (c ^ '0');}a = f ? -x : x;
}
template <typename T, typename ...Argv>
inline void read(T &a, Argv &...argv) {
    read(a), read(argv...);
}

int head[MAXn + 10], cntnex, nex[MAXm + 10], from[MAXm + 10], to[MAXm + 10];
inline void Insert(int u, int v) {
    nex[++cntnex] = head[u];
    head[u] = cntnex;
    from[cntnex] = u;
    to[cntnex] = v;
}

int n, W, ndw[MAXn + 10], ndv[MAXn + 10];
int cntdfs, idxdfs[MAXn + 10], bottom[MAXn + 10];
void Dfs(int cur) {
    idxdfs[++cntdfs] = cur;
    for (int i = head[cur]; i; i = nex[i]) {
        Dfs(to[i]);
    }
    bottom[cur] = cntdfs;
}

int d[MAXn * 2 + 10][MAXW + 10];
signed main() {
    read(n, W);
    for (int i = 1, f; i <= n; ++i) {
        read(ndw[i], f, ndv[i]);
        if (f == i) {
            Insert(n + 1, i);
        } else {
            Insert(f, i);
        }
    }
    Dfs(n + 1);
    // 主干部分
    for (int i = cntdfs; i; --i) {
        int cur = idxdfs[i];
        for (int j = W; j >= ndw[cur]; --j) {
            d[i][j] = max(d[bottom[cur] + 1][j], d[i + 1][j - ndw[cur]] + ndv[cur]);
        }
        for (int j = ndw[cur] - 1; ~j; --j) {
            d[i][j] = d[bottom[cur] + 1][j];
        }
    }
    // 主干部分
    int ans = 0;
    for (int j = 0; j <= W; ++j) {
        ans = max(ans, d[1][j]);
    }
    printf("%d\n", ans);
    return 0;
}