New Twiddle

components.tree-elem.js

import Ember from 'ember';

export default Ember.Component.extend({
  isSelected: Ember.computed('selectedDirs.[]', 'path', function() {
    return this.get('selectedDirs')
      .some(d => this.get('path').startsWith(d));
  })
});

components.tree-root.js

import Ember from 'ember';

export default Ember.Component.extend({
  init() {
    this._super(...arguments);
    this.set('selectedDirs', []);
  },
  actions: {
    toggle(elem) {
      
      if(!this.get('selectedDirs').includes(elem)) {
        this.set('selectedDirs', [elem, ...this.get('selectedDirs')]);
      } else {
        this.set('selectedDirs', this.get('selectedDirs').filter(x => x != elem));
      }
    }
  }
});

controllers.application.js

import Ember from 'ember';

export default Ember.Controller.extend({
  tree: {
    europe: {
      germany: { berlin: {}, munich: {} },
      france: { paris: {}, milano: {} }
    },
    africa: {}
  }
});

styles.app.css

body {
  margin: 12px 16px;
  font-family: 'Helvetica Neue', Helvetica, Arial, sans-serif;
  font-size: 12pt;
}

.dir {
  margin: .25rem;
  margin-left: 1rem;
  border: 2px solid black;
  padding: .25rem;
}

.dir.selected {
  border-color: blue;
}

templates.application.hbs

{{tree-root tree=tree}}

templates.components.tree-elem.hbs

<div class="dir {{if isSelected 'selected'}}">
    {{path}} <button onclick={{action toggle path}}>toggle</button>
    {{#each-in this.elem as |name children|}}
      {{tree-elem
        path=(concat path name '/')
        elem=children
        selectedDirs=selectedDirs
        toggle=toggle
      }}
    {{/each-in}}
</div>

templates.components.tree-root.hbs

Tree:
{{tree-elem
  path="/"
  elem=tree
  selectedDirs=selectedDirs
  toggle=(action 'toggle')
}}

Selected{{selectedDirs}}:
<ul>
{{#each selectedDirs as |dir|}}
  <li>{{dir}}</li>
{{/each}}
</ul>

twiddle.json

{
  "version": "0.15.1",
  "EmberENV": {
    "FEATURES": {}
  },
  "options": {
    "use_pods": false,
    "enable-testing": false
  },
  "dependencies": {
    "jquery": "https://cdnjs.cloudflare.com/ajax/libs/jquery/3.3.1/jquery.js",
    "ember": "3.4.3",
    "ember-template-compiler": "3.4.3",
    "ember-testing": "3.4.3"
  },
  "addons": {
    "ember-data": "3.4.2"
  }
}