Indeed, it works fine on a test page which have the only those controls. However, in real website scenario with 3rd party libraries like Bootstrap & custom css (which we have) that most likely won't work as the dropdown <div> with absolute positioning is created in the body container which might have different sizing to compare with the box sizing where the control is located so the width value in percentage format won't work.
Try this code
<%@ Page Language="C#" AutoEventWireup="true" %>
<!DOCTYPE html>
<head runat="server">
<style>
.rddtSlide {
width: 99% !important;
}
</style>
</head>
<body>
<form id="form1" runat="server">
<telerik:RadScriptManager runat="server"></telerik:RadScriptManager>
<div>
<h2>Works fine</h2>
<telerik:RadDropDownTree ID="rdroptreeEmployees" runat="server"
Width="25%" RenderMode="Classic"
DataFieldID="EmployeeID"
DataFieldParentID="ReportsTo"
DataSourceID="sqldsEmployees" DataTextField="FirstName">
</telerik:RadDropDownTree>
<telerik:RadDropDownTree ID="RadDropDownTree1" runat="server"
Width="25%" RenderMode="Lightweight"
DataFieldID="EmployeeID"
DataFieldParentID="ReportsTo"
DataSourceID="sqldsEmployees" DataTextField="FirstName">
</telerik:RadDropDownTree>
</div>
<div style="width: 600px;">
<h2>Does not work</h2>
<telerik:RadDropDownTree ID="RadDropDownTree2" runat="server"
Width="25%" RenderMode="Classic"
DataFieldID="EmployeeID"
DataFieldParentID="ReportsTo"
DataSourceID="sqldsEmployees" DataTextField="FirstName">
</telerik:RadDropDownTree>
<telerik:RadDropDownTree ID="RadDropDownTree3" runat="server"
Width="25%" RenderMode="Lightweight"
DataFieldID="EmployeeID"
DataFieldParentID="ReportsTo"
DataSourceID="sqldsEmployees" DataTextField="FirstName">
</telerik:RadDropDownTree>
</div>
</form>
</body>
</html>