test.js

function Node(cost) {
  this.cost = cost;
  this.children = [];
}

function getCheapestCost(rootNode) {
    var n = rootNode.children.length;
    
    // initialize minCost to the largest integer in the system
    var minCost = Number.MAX_SAFE_INTEGER;

    if (n === 0) {
        return rootNode.cost;
    } else {
        for(let i = 0; i < n; ++i) {
            let tempCost = getCheapestCost(rootNode.children[i]);
            if (tempCost < minCost)
                minCost = tempCost;
        }
    }
    return minCost + rootNode.cost
}

var root = new Node(0);
console.log(getCheapestCost(root));
root.children = [new Node(5), new Node(3), new Node(6)];
root.children[0].children = [new Node(4)];
root.children[1].children = [new Node(2), new Node(0)];
root.children[2].children = [new Node(1), new Node(5)];
root.children[1].children[0].children = [new Node(1)];
console.log(getCheapestCost(root));
root.children[1].children[1].children = [new Node(10)];
console.log(getCheapestCost(root));
root.children[1].children[0].children[0].children = [new Node(1)];
console.log(getCheapestCost(root));
root.children[1].children[0].children[0].children[0].children = [new Node(4)];
console.log(getCheapestCost(root));
root.children[2].children[0] = new Node(3);
console.log(getCheapestCost(root));