bfs.js

const dataset = [
  {
    "key":'a',
    "children": [
      {
        "key":'a1',
        "children": [
          {
            "key":'a1a',
            "children": [],
          },
          {
            "key":'a1b',
            "children": [],
          },
        ],
      },
      {
        "key":'a2',
        "children": [
          {
            "key":'a2a',
            "children":[
              {
                "key":'a2a1',
                "children": [],
              },
            ]
          },
          {
            "key":'a2b',
            "children": [],
          },
        ],
      },
      {
        "key":'a3',
        "children": [
          {
            "key":'a3a',
            "children": [],
          },
          {
            "key":'a3b',
            "children": [
              {
                "key":'a3b1',
                "children": [],
              },
            ],
          },
        ],
      },
    ],
  },
  {
    "key":'b',
    "children": [],
  },
];

function bfs(parentRoute, dataset, target) {
  const nextNodes = [];
  for (node of dataset) {
    const { key, children } = node;
    const route = parentRoute.slice();
    route.push(node.key);

    if (key === target) {
      return route;
    }

    if (children) {
      nextNodes.push(() => bfs(route, children, target));
    }
  } 

  for (next of nextNodes) {
    const route = next();
    if (route !== undefined) {
      return route;
    }
  }
}

const routeTaken = bfs([], dataset, 'a3b1');
console.log(routeTaken);