Workaround 1;

input[type=text]::-ms-clear {
    display: none;
}

Workaround 2: 

// https://www.telerik.com/support/kb/aspnet-ajax/details/how-to-create-a-javascript-array-of-all-particular-radcontrols-on-the-page
function get_allRadDropDownTrees() {
    var allRadDropDownTrees = [];
    var allRadControls = $telerik.radControls;

    for (var i = 0; i < allRadControls.length; i++) {
        var element = allRadControls[i];

        if (Telerik.Web.UI.RadDropDownTree && element instanceof Telerik.Web.UI.RadDropDownTree) {
            Array.add(allRadDropDownTrees, element);
        }
    }
    return allRadDropDownTrees;
}


function fixClearButton(sender, args) {
    tree = sender;
    if ($telerik.isIE) {
        var dropDownTrees = get_allRadDropDownTrees();
        dropDownTrees.forEach(function (dropdowntree) {                       
            $telerik.$(dropdowntree.get_filterElement()).on('input', function (e, a) {
                if (!e.target.value) {
                    dropdowntree.filterByText("")
                }
            })
        })
    }
    Sys.Application.remove_load(fixClearButton);
}

Sys.Application.add_load(fixClearButton);



Workaround 3: 

var tree;
function OnClientLoad(sender, args) {
    tree = sender;
    if ($telerik.isIE) {
        $telerik.$(sender.get_filterElement()).on('input', function (e, a) {
            if (!e.target.value) {
                tree.filterByText("")
            }
        })
    }
}

Reproduction: https://www.screencast.com/t/Y6HRTmGngkJ

In order to workaround this bug, you can handle the OnClientNodeClicking client-side event of the embeddedTree and cancel the propagation of the event if the node that is about to be selected is disabled:

 protected void Page_Load(object sender, EventArgs e)
 {
        Reporting.EmbeddedTree.OnClientNodeClicking = "OnClientNodeClicking";
 }

<script type="text/javascript">
            function OnClientNodeClicking(sender, args) {
                var nodeToSelect = args.get_node();

                if (!nodeToSelect.get_enabled())
                    args.set_cancel(true);
            }
        </script>

The DropDownTree pop-up is not positioned properly when the <form> is wrapped inside a relatively positioned element with fixed width.

The issue can be reproduced with a similar configuration:

    <div style="width: 700px; position: relative; left:300px">
        <form id="form1" runat="server">
            <telerik:RadScriptManager runat="server"></telerik:RadScriptManager>
            <div class="position">
                <telerik:RadDropDownTree runat="server" ID="ddt1" CssClass="dropDownTree"></telerik:RadDropDownTree>
            </div>
        </form>
    </div>

When I set the RadDropDownTree Enbaled as False, the Clear button still working and clear the Selected value.

Validation is not working as expected in Dropdowntree. In the demo page http://demos.telerik.com/aspnet-ajax/dropdowntree/examples/validation/defaultcs.aspx, if any one node is checked, the validation message "You need to select a type or a category before submitting the page" is displayed. If multiple checkboxes are checked, the message is not displayed. I do not want the validation message displayed when  check box is  checked.