JSFiddle - React, Tailwind, and code Playground

HTML

Enter search terms: <input id="search-input" />

<ul class="search-tree">
    <li>
        <span>Root node</span>
        <ul>
            <li>
                <span>One Child Node</span>
                <ul>
                    <li><span>node 1</span></li>
                    <li><span>node 2</span></li>
                    <li><span>node 3</span></li>
                </ul>
            </li>
            <li>
                <span>Two Child Node</span>
                <ul>
                    <li><span>node 4</span></li>
                    <li><span>node 5</span></li>
                    <li><span>node 6</span></li>
                </ul>
            </li>
        </ul>
    </li>
</ul>

JavaScript

function process(li, path) {
  
    $.data($(li).get(0), 'path', path)

    $(li).children("ul").each(function(jdx, ul) {
         $(ul).children("li").each(function(idx, child_li) {
             process($(child_li), path + " / " + $(child_li).children("span").text())
         });
    }); 
}

$(".search-tree li").each(function(idx, li) {
    process($(li), $(li).children("span").text())    
});

$("#search-input").keyup(function() {
    // Hide everything to start with
    $(".search-tree li").hide()
    
    // Extract all the words from the search box
    var words = $("#search-input").val().split(" ");
    
    // Now go through the tree, and show <li>s that match the words
    $(".search-tree").find("li").each(function(idx, li) {
        // We only test the nodes, ie, the <li>s with no children
        if ($(li).find("ul").length == 0) {
            // Assume we will show the node...
            var show = true;
            // ...but decide to hide it if one of the words doesn't exist in the path
            $(words).each(function(jdx, word) {
              
                if ($.data($(li).get(0), 'path').indexOf(word) == -1) {
                    show = false
                }
            })
            // If the verdict was show=true, show this node and its parents
            if (show) {
                $(li).show()
                $(li).parents("li").show()
            }
        }
    });
});