I need some help with os.walk and defining shapefiles in sub folders

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])