SOURCE

var Main = {
    watch: {
      filterText(val) {
        if (val) {
          this.$refs.tree.filter(val);
        } else {
          this.keyTree = + new Date()
        };
      }
    },

    methods: {
      filterNode(value, data) {
        if (!value) {
          return true
        };
        return data.label.indexOf(value) !== -1;
      }
    },

    data() {
      return {
        keyTree: + new Date(),
        filterText: '',
        data: [{
          id: 1,
          label: '一级 1',
          children: [{
            id: 4,
            label: '二级 1-1',
            children: [{
              id: 9,
              label: '三级 1-1-1'
            }, {
              id: 10,
              label: '三级 1-1-2'
            }]
          }]
        }, {
          id: 2,
          label: '一级 2',
          children: [{
            id: 5,
            label: '二级 2-1'
          }, {
            id: 6,
            label: '二级 2-2'
          }]
        }, {
          id: 3,
          label: '一级 3',
          children: [{
            id: 7,
            label: '二级 3-1'
          }, {
            id: 8,
            label: '二级 3-2'
          }]
        }],
        defaultProps: {
          children: 'children',
          label: 'label'
        }
      };
    }
  };
var Ctor = Vue.extend(Main)
new Ctor().$mount('#app')

<script src="//unpkg.com/vue@2/dist/vue.js"></script>
<script src="//unpkg.com/element-ui@2.15.14/lib/index.js"></script>
<div id="app">
<el-input
  placeholder="输入关键字进行过滤"
  v-model="filterText">
</el-input>

<el-tree
  class="filter-tree"
  :data="data"
  :props="defaultProps"
  :filter-node-method="filterNode"
  :key="keyTree"
  ref="tree">
</el-tree>
</div>

@import url("//unpkg.com/element-ui@2.15.14/lib/theme-chalk/index.css");