So far this is what I have. It works if the shapefiles are in the C:/test_define folder, but it will fail with this error if they are in sub 
folders. I am new to this so I am probably missing something simple, but I can't figure it out.. 
2000-10-Straight hole well.shp
--------------------------------------------------
Failed to execute. Parameters are not valid.
ERROR 000732: Input Dataset or Feature Class: Dataset 2000-10-Straight hole well.shp does not exist or is not supported
Failed to execute (DefineProjection).
My code
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
import os
import arcpy
# Set workspace environment to a folder
WS = "c:/test_define"
arcpy.env.workspace = WS
for (path, dirs, files) in os.walk(WS):
    for name in files:
        if name.endswith(".shp"):
            print name
            print ("-") * 50
            try:
                infc = name
                sr = arcpy.SpatialReference("NAD 1983 UTM Zone 11N")
                arcpy.DefineProjection_management(infc, sr)
 
                # print messages when the tool runs successfully
                print(arcpy.GetMessages(0))
    
            except arcpy.ExecuteError:
                print(arcpy.GetMessages(2))
    
            except Exception as ex:
                print(ex.args[0])