<el-tree 
ref="menuTree" 
:data="menuTreeList" 
:check-strictly="true" 
show-checkbox 
node-key="id" 
highlight-current
:filter-node-method="filterNode"
@node-click="nodeClick"  
@check-change="handleCheckChange" 
default-expand-all/></el-tree>

//由于nodeClick被调用时checkChange也会被调用，因此需要一个editCheckId辅助避免出现因为联动导致选中状态错乱
            nodeClick(data, node, v) {
                this.editCheckd = data.id
                //this.$refs.menuTree.setCheckedKeys([data.id]);
                this.menu.parentId = data.parentId || null;
                this.menu.productId = data.id;
                this.menu.typeName = data.typeName;
                this.menu.typeCode = data.typeCode;
                this.menu.status = data.status;
                this.menu.sort = data.sort;
                this.$refs.form.clearValidate();
            },

handleCheckChange(item,checked,self){
                if(checked){
                    this.editCheckd = item.id
                    this.$refs.menuTree.setCheckedKeys([item.id]);
                    // this.nodeClick(item)
                }else{
                    if(this.editCheckd === item.id){
                        this.$refs.menuTree.setCheckedKeys([item.id]);
                    }
                    
                }
            },

查阅链接：

Vue单页应用中Element ui中的el-tree单选功能 - 简书