node_type *refresh(node_type *p) const {
if(p == nullptr){ return nullptr; }
value_type cache = p->value;
if(p->children[0] != nullptr){
cache = m_traits(p->children[0]->cache, cache);
}
if(p->children[1] != nullptr){
cache = m_traits(cache, p->children[1]->cache);
}
p->size = 1 + size(p->children[0]) + size(p->children[1]);
p->cache = m_traits.resolve(p->size, cache, p->modifier);
return p;
}
void propagate(node_type *p) const {
if(p == nullptr){ return; }
const size_t k = size(p->children[0]);
const auto mod_l_cr = m_traits.split_modifier(p->modifier, k);
const auto mod_c_r = m_traits.split_modifier(mod_l_cr.second, 1);
if(p->children[0] != nullptr){
p->children[0]->modifier = m_traits.merge_modifier(
p->children[0]->modifier, mod_l_cr.first);
refresh(p->children[0]);
}
if(p->children[1] != nullptr){
p->children[1]->modifier = m_traits.merge_modifier(
p->children[1]->modifier, mod_c_r.second);
refresh(p->children[1]);
}
p->value = m_traits.resolve(1, p->value, mod_c_r.first);
p->modifier= m_traits.default_modifier();
}