It's not advisable to chop down trees with a flimsy axe as the axe is likely to break and you could end up injuring yourself. If you're determined to use a flimsy axe, then make sure you take extra care and go slowly. Start by chopping small branches off the tree before working your way up to larger ones. Use a wedge to help split the wood if necessary. Be extra careful when cutting through the trunk of the tree as it's under a lot of pressure. Again, use a wedge if needed and take your time. Once the tree starts to lean, stop chopping and get out of the way quickly in case it falls on you.