Are there any efficient ways to populate a balanced tree structure

问题

I have a balanced binary tree structure:

Node 0 at depth 0 is the root. The root's left child is 1 and right child is 2, and so on.

Please see image:

The total depth of the tree is given as N. This N is the only parameter of the problem. Nodes at level N are designated as leaf nodes.

I am storing this tree using the following node structure.

struct node_s{
    int n, depth, parent;//n is node number
    int nodescendents;//number of descendents of the current node
    std::vector<int> descendents;//Descendents in ascending order
    int lchild, rchild;//Immediate left child and right child
    std::vector<int> lchildleaves;//leaf nodes that descend from the immediate 
                                                      //left child
    std::vector<int> rchildleaves;//leaf nodes that descend from the immediate 
                                                      //right child
};

I intend to store the tree itself as:

std::vector<node_s> tree;

Is there a way to numerically efficiently populate the tree vector using simple algebra roughly as:

//Creating the nth node, beginning from 0th node, then 1st node and so on
nodes_s node;
//populate all data structures of the nth node
//precisely, here, there are loops, algebraic calculations, etc., that can help 
//populate all of the node_s data members.
tree.push_back(node);

The only way I can think of as of now is to construct a graph explicitly and run some sort of Dijkstra algorithm to figure out these data structure values for each node.

回答1:

For a node k, the key point is to identify its position in the graph, in order to identify its parent, if it is a left or right child.

For node k, its rank r[k] is equal to floor(log2(k+1)) and its position in the rank is equal to p[k] = k - 2^r[k] + 1

Then k is located by the pair (r[k], p[k])

Conversely, k = 2^r[k] + p[k] - 1

Its parent is then located by (r[k]-1, floor(p[k]/2)) -> node index = 2^r + p - 1

k is a left child if k%2 == 1

I guess the rest is rather easy

来源：https://stackoverflow.com/questions/53349791/are-there-any-efficient-ways-to-populate-a-balanced-tree-structure