Functional style oriented graph traversal https://jsfiddle.net/tripolskypetr/132am0bL/

orientedTree.js

//            o      <- end
//           / \   n
//      f   g   \ /
//       \ /     m  n
//        e      \ /
//        |    p  |
//     h  d     \ /
//      \ |      l
//     i  c     /  
//      \ |    k
//        b   /
//         \ /
//          a <- start


// This is a directed graph. Accordingly, we represent it in the form of pairs of points

    const roads = {
    	a: ['b', 'k'],
    	b: ['c'],
    	c: ['i', 'd'],
    	d: ['h', 'e'],
    	e: ['f', 'g', 'o'],
        f: [],
        g: [],
        h: [],
    	i: [],
        k: ['l'],
        l: ['m', 'p'],
        m: ['o', 'n'],
        n: [],
        o: [],
        p: [],
    };

// We write a function that will iterate over all possible roads, adjusted for the fact that two different points can lead to the same endpoint (applyWay). We also write a function that recursively collects graph connections into segments (buildWays) the start and end points.

    const buildWays = (dt) => [...roads[dt].map((d) => [dt, d]), ...roads[dt].map((d) => buildWays(d)).reduce((a, v) => a.concat(v), [])];

    const first = (arr) => arr[0];
    const last = (arr) => arr[arr.length - 1];
    
    const applyWay = (dot) => {
    	const apply = [];
    	ways.forEach((way) => {
    		if (last(way) === dot) {
                roads[dot].forEach((road) => apply.push([...way, ...road]));
            } else {
    			return;
            }
        });
    	ways.push(...apply);
    }

// Count all kinds of segments. Using the start and end points we connect the segments in various routes

    const ways = buildWays('a'); // [ ['a','b'], ['b','c'] ]
    Object.keys(roads).forEach((dot) => applyWay(dot)/* [ ['a','b', 'c', 'i'], ... ] */);

// From the start and end points we find the shortest route

    const calculateRoute = (to, from = 'a') => ways
      .filter((way) => first(way) === from && last(way) === to)
      .sort((a, b) => a.length > b.length ? 1 : a.length < b.length ? -1 : 0);

    console.log(first(calculateRoute('o'))); // [ 'a', 'k', 'l', 'm', 'o' ]