probably wrote:

function compare(value, keyword){
    if(value === keyword){
        return 2;
    }
    if(new RegExp('^' + value).test(keyword)){
        return 1;
    }
    return 0;
}

function walk(node, keyword){

    let isFind = false;
    const path = [];
    let children = null;

    function dfs(node, keyword){
        if(compare(node.value, keyword) === 1){
            path.push(node.value);
            node.children && node.children.forEach(childNode => {
                dfs(childNode, keyword);
            });
        }
        if(compare(node.value, keyword) === 2) {
            node.children && (children = node.children);
            isFind = true;
        }
    }

    dfs(node, keyword);

    return {
        isFind,
        path,
        children
    }
}

function generateObj(pathArr, keyword, children){
    const resObj = {};
    let tempObj = resObj;
    pathArr.forEach(path => {
        tempObj.value = path;
        tempObj.children = [{}];
        tempObj = tempObj.children[0];
    });
    tempObj.value = keyword;
    children && (tempObj.children = children);
    return resObj;
}

function search(nodes, keyword){
    const searchedNodes = [];
    nodes.forEach(node => {
        const { isFind = false, path, children = null } = walk(node, keyword);
        if(isFind){
            searchedNodes.push(generateObj(path, keyword, children));
        }
    });
    return searchedNodes;
}

const nodes = [
  {
    value: "1-1",
    children: [
      { value: "1-1-1"},
      { value: "1-1-2", children:[
        {
          value: "1-1-2-1",
          children: [
            { value: "1-1-2-1-1" },
            { value: "1-1-2-1-2" }
          ]
        },
        {
          value: "1-1-2-2"
        }
      ] }
    ]
  },
  {
    value: "1-2",
    children: [
      { value: "1-2-1"},
      { value: "1-2-2", children:[
        {
          value: "1-2-2-1",
          children: [
            { value: "1-2-2-1-1" },
            { value: "1-2-2-1-2" }
          ]
        },
        {
          value: "1-2-2-2"
        }
      ] }
    ]
  },
];

const searchedNodes = search(nodes, "1-1-2-1"); 

the expected data and problem description you gave are ambiguous,

  children: [
            { value: "1-1-2-1-1" },
            { value: "1-1-2-1-2" }
          ]

is also consistent with 1-1-2-1 .