DFS output all matching elements and its ancestor : javascript - recursion

Question

Given
// tree data structure
Elephant --> Duck
--> Hamster --> Elephant1
--> Elephant2
Hamster --> Elephant --> Fish
--> Dog -------> Elephant
Dog --> Unicorn
--> Fish --> Hamster
--> Elephant --> Elephant
when user type term 'Ele', would like to output result so that all the tree is reduced to the following format (which has the matching result with its ancestor)
Elephant --> Hamster --> Elephant1
--> Elephant2
Hamster --> Elephant
--> Dog -------> Elephant
Dog --> Fish --> Elephant --> Elephant
Given
this.tree = [
{ id:1, name: 'Elephant',children:[
{ id:2, name: 'Duck' },
{ id:3, name: 'Hamster', children: [
{ id: 4, name: 'Elephant1', id: 5, name: 'Elephant2' }]
}
]},
{ id:5A, name: 'Hamster', children: [
{ id:6, name: 'Elephant', children: [
{ id:7, name: 'Fish' }
]},
{ id:8, name: 'Dog', children: [
{ id:9, name: 'Elephant' }
]}
]},
{ id:10, name: 'Dog', children: [
{ id:11, name: 'Unicorn' },
{ id:12, name: 'Fish', children: [
{ id:13, name: 'Hamster' },
{ id:14, name: 'Elephant', children:
[{ id:15, name: 'Elephant' }
]},
]}
]},
{ id:16, name: 'Elephant', children: [
{ id:17, name: 'Duck' },
{ id:18, name: 'Hamster', children: [
{ id:19, name: 'Elephant' },
{ id:20, name: 'Fish' }
]}
]}
]
My attempt:
I have tried the following using Depth First Search traversal but having issue to include its ancestor when its children has matching term. Currently the code only prints the matching result of the children but not its ancestor. I am half way through the solution (probably need some slight modification on my attempt solution) but is having trouble in the backtracking. Have been searching for resources on DFS and Would appreciate if anybody know about this.
onSearch(term) {
let found = this.search(term, JSON.parse(JSON.stringify(this.tree)));
console.log(found);
}
search(term, parent, path, basket) {
let fork = path.slice(0);
let found, { name, id, children } = parent;
let nameId = name + ':' +id;
fork.push(nameId);
if (name.toLowerCase().indexOf(term) !== -1) {
basket.push(fork);
fork = [];
return basket
}
if (!!children && children.length > 0) {
for (let child of parent.children) {
this.search(term, child, fork, basket)
}
return basket;
}
if (children.length === 0) {
return [];
}
}
this.onSearch('elep');
However the above attempt solution return the below result which is not exactly right.
Elephant --> Hamster --> Elephant1
Hamster --> Elephant
--> Dog -------> Elephant
Dog --> Fish --> Elephant
Desire solution is to output
Elephant --> Hamster --> Elephant1
--> Elephant2
Hamster --> Elephant
--> Dog -------> Elephant
Dog --> Fish --> Elephant --> Elephant

Answer 1

After few attempts, found the following solution which will output the desire solution
onSearch(term) {
let found = this.search(term, JSON.parse(JSON.stringify(this.tree)));
console.log(found);
}
search(term, parent, path, basket) {
let found, { name, id, children } = parent;
let nameId = name + ':' +id;
let fork = path.slice(0);
fork.push(nameId);
if (name.toLowerCase().indexOf(term) !== -1) {
basket.push([fork]);
}
if (!!children && children.length > 0) {
for (let child of parent.children) {
this.search(term, child, fork, basket)
}
return basket;
}
}