C# code to remove the xml node that doesnt have value

I have a below xml file. I need to read the file and remove the book which doesn't have price mentioned in it and save it back. The new file should have only the books which has prices in it.

<catalog>
  <book>
    <author>Gambardella, Matthew</author>
    <title>XML Developer's Guide</title>
    <genre>Computer</genre>
    <price>44.95</price>
    <publish_date>2000-10-01</publish_date>
    <description>
      An in-depth look at creating applications
      with XML.
    </description>
  </book>
  <book>
    <author>Ralls, Kim</author>
    <title>Midnight Rain</title>
    <genre>Fantasy</genre>
    <price>5.95</price>
    <publish_date>2000-12-16</publish_date>
    <description>
      A former architect battles corporate zombies,
      an evil sorceress, and her own childhood to become queen
      of the world.
    </description>
  </book>
  <book>
    <author>Corets, Eva</author>
    <title>Maeve Ascendant</title>
    <genre>Fantasy</genre>
    <price/>
    <publish_date>2000-11-17</publish_date>
    <description>
      After the collapse of a nanotechnology
      society in England, the young survivors lay the
      foundation for a new society.
    </description>
  </book>
  <book>
    <author>Corets, Eva</author>
    <title>Oberon's Legacy</title>
    <genre>Fantasy</genre>
    <price>5.95</price>
    <publish_date>2001-03-10</publish_date>
    <description>
      In post-apocalypse England, the mysterious
      agent known only as Oberon helps to create a new life
      for the inhabitants of London. Sequel to Maeve
      Ascendant.
    </description>
  </book>
  <book>
    <author>Corets, Eva</author>
    <title>The Sundered Grail</title>
    <genre>Fantasy</genre>
    <price/>
    <publish_date>2001-09-10</publish_date>
    <description>
      The two daughters of Maeve, half-sisters,
      battle one another for control of England. Sequel to
      Oberon's Legacy.
    </description>
  </book>
  <book>
    <author>Randall, Cynthia</author>
    <title>Lover Birds</title>
    <genre>Romance</genre>
    <price>4.95</price>
    <publish_date>2000-09-02</publish_date>
    <description>
      When Carla meets Paul at an ornithology
      conference, tempers fly as feathers get ruffled.
    </description>
  </book>
  <book>
    <author>Thurman, Paula</author>
    <title>Splish Splash</title>
    <genre>Romance</genre>
    <price>4.95</price>
    <publish_date>2000-11-02</publish_date>
    <description>
      A deep sea diver finds true love twenty
      thousand leagues beneath the sea.
    </description>
  </book>
  <book>
    <author>Knorr, Stefan</author>
    <title>Creepy Crawlies</title>
    <genre>Horror</genre>
    <price>4.95</price>
    <publish_date>2000-12-06</publish_date>
    <description>
      An anthology of horror stories about roaches,
      centipedes, scorpions  and other insects.
    </description>
  </book>
  <book>
    <author>Kress, Peter</author>
    <title>Paradox Lost</title>
    <genre>Science Fiction</genre>
    <price/>
    <publish_date>2000-11-02</publish_date>
    <description>
      After an inadvertant trip through a Heisenberg
      Uncertainty Device, James Salway discovers the problems
      of being quantum.
    </description>
  </book>
  <book>
    <author>O'Brien, Tim</author>
    <title>Microsoft .NET: The Programming Bible</title>
    <genre>Computer</genre>
    <price/>
    <publish_date>2000-12-09</publish_date>
    <description>
      Microsoft's .NET initiative is explored in
      detail in this deep programmer's reference.
    </description>
  </book>
  <book>
    <author>O'Brien, Tim</author>
    <title>MSXML3: A Comprehensive Guide</title>
    <genre>Computer</genre>
    <price>36.95</price>
    <publish_date>2000-12-01</publish_date>
    <description>
      The Microsoft MSXML3 parser is covered in
      detail, with attention to XML DOM interfaces, XSLT processing,
      SAX and more.
    </description>
  </book>
  <book>
    <author>Galos, Mike</author>
    <title>Visual Studio 7: A Comprehensive Guide</title>
    <genre>Computer</genre>
    <price/>
    <publish_date>2001-04-16</publish_date>
    <description>
      Microsoft Visual Studio 7 is explored in depth,
      looking at how Visual Basic, Visual C++, C#, and ASP+ are
      integrated into a comprehensive development
      environment.
    </description>
  </book>
</catalog>

Please help me to write a c# code to do the same.